# y=x3,y=xy=x3,y=x Sketch the region enclosed by the given curves and find its area.

**SCIENCE**

**True or false? Boiling point and melting point are physical properties that can be used to…**

The boiling point is the temperature at which a substance converts from liquid phase to gas phase. The melting point, on the other hand, is the temperature at which a substance converts from solid…

1 educator answer

**MATH**

**π(∫π20((1+cos(x))2−12)dx)π(∫0π2((1+cos(x))2-12)dx) Each integral represents the volume of a solid….**

The formula provided represents the volume of the solid obtained by rotating the region enclosed by the curves y=1+cosx,y=1,x=0,x=π2y=1+cosx,y=1,x=0,x=π2 , about y axis, using washer method:

1 educator answer

**MATH**

**π(∫10(y4−y8)dy)π(∫01(y4-y8)dy) Each integral represents the volume of a solid. Describe the solid.**

The formula provided represents the volume of the solid obtained by the rotation of the region bounded by the curves x=y2,x=y4x=y2,x=y4 , the lines x = 0, x = 1, about y axis.

1 educator answer

**MATH**

**π(∫1−1(1−y2)2dy)π(∫-11(1-y2)2dy) Each integral represents the volume of a solid. Describe the…**

The formula provided represents the volume of the solid obtained by rotating the region enclosed by the curves x=y2,x=1x=y2,x=1 , about x = 1, using washer method:

1 educator answer

**MATH**

**π(∫π0(sin(x))dx)π(∫0π(sin(x))dx) Each integral represents the volume of a solid. Describe the solid.**

The formula provided represents the volume of the solid obtained by rotating the region enclosed by the curves y=sinx−−−−√,y=0,y=sinx,y=0, about y axis, using washer method:

1 educator answer

**MATH**

**y=x,y=xe1−(x2)y=x,y=xe1-(x2) Use a computer algebra system to find the exact volume of the…**

These curves are intersected at x=0x=0 and x=2.x=2. Between these points 0≤x≤xe1−x2≤2.0≤x≤xe1-x2≤2. Let’s use the method of rings. The parameter of a ring is xx between 00 and 2.2. The area of a…

1 educator answer

**MATH**

Let’s use the method of rings. The parameter for a ring will be xx from x=0x=0 to x=π.x=π. The area of a ring is π((1+sin2(x))2−12)=π(sin4(x)+2sin2(x))π((1+sin2(x))2-12)=π(sin4(x)+2sin2(x)) (here 11 is the inner radius of the…

1 educator answer

**MATH**

**y=3sin(x2),y=ex2+e−2xy=3sin(x2),y=ex2+e-2x Use a graph to find approximate x-coordinates of the…**

We can see the two graphs intersect at about x = .75 and about x = 1.5. Now we want to set up an integral where we rotate this area around the x axis and make washers. The formula for washers…

1 educator answer

**MATH**

**y=2+(x2)cos(x),y=x4+x+1y=2+(x2)cos(x),y=x4+x+1 Use a graph to find approximate x-coordinates of the…**

y=2+x2cos(x),y=x4+x+1y=2+x2cos(x),y=x4+x+1 Refer the attached image. Graph of y=2+x2cos(x)y=2+x2cos(x) is plotted in red color and graph of y=x4+x+1y=x4+x+1 is plotted in blue color. From the graph the curves intersect at…

1 educator answer

**MATH**

**y=x,y=0,x=2,x=4y=x,y=0,x=2,x=4 Find the volume of the solid obtained by rotating the region…**

You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves y=x,y=0,x=2,x=4y=x,y=0,x=2,x=4about x = 1, using washer method, such that:

1 educator answer

**MATH**

**x=y2,x=1−y2x=y2,x=1-y2 Find the volume of the solid obtained by rotating the region bounded…**

You need to evaluate the volume using the washer method, such that: V=π⋅∫ba(f2(x)−g2(x))dxV=π⋅∫ab(f2(x)-g2(x))dx You need first to determine the endpoints, hence you need to solve for y the following…

1 educator answer

**MATH**

**xy=1,y=0,x=1,x=2xy=1,y=0,x=1,x=2 Find the volume of the solid obtained by rotating the region…**

By using Washer method, we can find the volume of the solid. V=π⋅∫ba(f2(y)−g2(y)dyf(y)>g(y)V=π⋅∫ab(f2(y)-g2(y)dyf(y)>g(y) Since the curve is bounded by x=1 and x=2, then y-values are bounded between 0 and…

1 educator answer

**MATH**

**y=x3,y=0,x=1y=x3,y=0,x=1 Find the volume of the solid obtained by rotating the region bounded…**

You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves y=x3,y=0,x=1y=x3,y=0,x=1 , about x = 2, using washer method, such that:

1 educator answer

**MATH**

**y=sin(x),y=cos(x),0≤x≤(π4)y=sin(x),y=cos(x),0≤x≤(π4) Find the volume of the solid obtained by…**

We use Washer method to evaluate the volume of the solid. V=∫baπf2(x)−g2(x)dx,f(x)>g(x)V=∫abπf2(x)-g2(x)dx,f(x)>g(x) Here f(x)=sinx,g(x)=cosxf(x)=sinx,g(x)=cosx x=0andπ4x=0andπ4 about y=−1y=-1

1 educator answer

**MATH**

**y=1+sec(x),y=3y=1+sec(x),y=3 Find the volume of the solid obtained by rotating the region bounded…**

y=1+sec(x),y=3y=1+sec(x),y=3 Refer the image. From the graph, the curves intersects at x=-pi/3 and x=pi/3. Using washer method, A cross section is a washer of cross sectional area A(x) with, Inner…

1 educator answer

**MATH**

**y=e−x,y=1,x=2y=e-x,y=1,x=2 Find the volume of the solid obtained by rotating the region…**

You need to find the volume of the solid obtained by rotating the region enclosed by the curves y=e−x,y=1,x=2y=e-x,y=1,x=2 , about y = 2, using washer method:

1 educator answer

**MATH**

**y=x2,x=y2y=x2,x=y2 Find the volume of the solid obtained by rotating the region bounded by…**

The volume of the solid obtained by rotating the region bounded by the curves y2=xy2=x and y=x2y=x2 about y = 1, can be evaluated using the washer method, such that:

1 educator answer

**MATH**

**y=(14)x2,x=2,y=0y=(14)x2,x=2,y=0 Find the volume of the solid obtained by rotating the region…**

You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves y=(14)x2,x=2y=(14)x2,x=2 , the line y = 0, about y axis, using washer method, such that:

1 educator answer

**MATH**

**y2=x,x=2yy2=x,x=2y Find the volume of the solid obtained by rotating the region bounded by…**

The volume of the solid obtained by rotating the region bounded by the curves y^2=x and x=2y about y axis, can be evaluated using the washer method, such that: V=∫baπ⋅(f2(x)−g2(x))dxV=∫abπ⋅(f2(x)-g2(x))dx…

1 educator answer

**MATH**

**y=(14)x2,y=5−x2y=(14)x2,y=5-x2 Find the volume of the solid obtained by rotating the region…**

y=x24,y=5−x2y=x24,y=5-x2 Refer the attached image. Graph of y=x24y=x24 is plotted in red color and graph of y=5−x2y=5-x2 is plotted in blue color. From the graph, the curves intersects at x=-2 , x=2. Using…

1 educator answer

**MATH**

**y=x3,y=x,x⇒0y=x3,y=x,x⇒0 Find the volume of the solid obtained by rotating the region…**

You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves y=x3,y=x,x=0y=x3,y=x,x=0 , about x axis, using washer method, such that:

1 educator answer

**MATH**

**y=ln(x),y=1,y=2,x=0y=ln(x),y=1,y=2,x=0 Find the volume of the solid obtained by rotating the…**

You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves y=lnx,y=1,y=2,x=0y=lnx,y=1,y=2,x=0about y axis, using washer method, such that:

1 educator answer

**MATH**

**x=2y√,x=0,y=9x=2y,x=0,y=9 Find the volume of the solid obtained by rotating the region…**

You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves x=2y√,x=0x=2y,x=0 , the line y = 9, about y axis, using washer method, such that:

1 educator answer

**MATH**

**y=25−x2−−−−−−√,y=0,x=2,x=4y=25-x2,y=0,x=2,x=4 Find the volume of the solid obtained by rotating…**

The volume of the solid obtained by rotating the region bounded by the curves y=25−x2−−−−−−√,y=0,x=2,x=4y=25-x2,y=0,x=2,x=4 , about x axis, can be evaluated using the washer method.

1 educator answer

**MATH**

**y=x−1−−−−−√,y=0,x=5y=x-1,y=0,x=5 Find the volume of the solid obtained by rotating the region…**

The volume of the solid obtained by rotating the region bounded by the curves y=x−1−−−−−√,y=0,x=5y=x-1,y=0,x=5 , about x axis, can be evaluated using the washer method, such that:

1 educator answer

**MATH**

**y=1−x2,y=0y=1-x2,y=0 Find the volume of the solid obtained by rotating the region bounded by…**

You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves y=1−x2,y=0y=1-x2,y=0 , about x axis, using washer method, such that:

1 educator answer

**MATH**

**y=2−(12)x,y=0,x=1,x=2y=2-(12)x,y=0,x=1,x=2 Find the volume of the solid obtained by rotating the…**

The volume of the solid obtained by rotating the region bounded by the curves y=2−x2,y=0,x=1,x=2y=2-x2,y=0,x=1,x=2 , about x axis, can be evaluated using the washer method, such that:

1 educator answer

**MATH**

To graph the area between curves, we determine the boundary values based on the intersection points. In the xy-plane, the graph of y=cos(x)y=cos(x) is plotted in red color while the graph of…

1 educator answer

**MATH**

**y=tan2(x),y=x−−√y=tan2(x),y=x Graph the region between the curves and use your calculator to…**

y=tan2(x),y=x−−√y=tan2(x),y=x Refer the attached image. The graph of y=tan2(x)y=tan2(x) is plotted in red color and the graph of y=x−−√y=x is plotted in blue color. From the graph the curves intersect a…

1 educator answer

**MATH**

**y=e1−x2,y=x4y=e1-x2,y=x4 Graph the region between the curves and use your calculator to…**

y=e1−x2,y=x4y=e1-x2,y=x4 Refer the attached image. Graph of y=e1−x2y=e1-x2 is plotted in red color and graph of y=x4y=x4 is plotted in blue color. From the graph,the given curves intersect at x=-1 and…

1 educator answer

**MATH**

**y=21+x4,y=x2y=21+x4,y=x2 Graph the region between the curves and use your calculator to…**

y=21+x4,y=x2y=21+x4,y=x2 Refer the attached image. Graph of y=21+x4y=21+x4is plotted in red color and graph of y=x2y=x2 is plotted in blue color. From the graph ,the curves intersect at x=-1 and x=1….

1 educator answer

**MATH**

**y=ex,y=2−x2y=ex,y=2-x2 Use a graph to find approximate x-coordinates of the points of…**

y=ex,y=2−x2y=ex,y=2-x2 Refer the attached image. Graph of e^x is plotted in red color and graph of y=2-x^2 is plotted in blue color. From the graph, the x-coordinates of the intersection of the curves…

1 educator answer

**MATH**

**y=3×2−2x,y=x3−3x+4y=3×2-2x,y=x3-3x+4 Use a graph to find approximate x-coordinates of the…**

y=3×2−2x,y=x3−3x+4y=3×2-2x,y=x3-3x+4 Refer the attached image. Graph of y=3×2−2xy=3×2-2x is plotted in red color and graph of y=x3−3x+4y=x3-3x+4is plotted in blue color. From the graph, the x-coordinates of the…

1 educator answer

**MATH**

**y=x(x2+1)2,y=x(x2+1)2, y=x5−x,y=x5-x, x≥0x≥0 Use a graph to find approximate x-coordinates of the…**

We can see that these graphs intersect at x = 0, and around x = 1. A little greater than x = 1. In order to approximate the area of this region we take the integral of the top function minus the…

1 educator answer

**MATH**

**y=xsin(x2),y=x4y=xsin(x2),y=x4 Use a graph to find approximate x-coordinates of the points of…**

The red curve refers to the graph of the first function: y=xsin(x2)y=xsin(x2)while the the the blue curve refers to the graph of the second function: y=x4y=x4 . As shown in the xy-plane, the two…

1 educator answer

**MATH**

**∫1−1(|3x−2x|)dx∫-11(|3x-2x|)dx Evaluate the integral and interpret it as the area of a region….**

Since 3x>2x3x>2x for x>0x>0 and 3x<2x3x<2x for x<0x<0 we have ∫1−1|3x−2x|dx=∫0−1(2x−3x)dx+∫10(3x−2x)dx=∫-11|3x-2x|dx=∫-10(2x-3x)dx+∫01(3x-2x)dx= Now we apply formula ∫ax=axlna+C.∫ax=axlna+C.

2 educator answers

**MATH**

∫π20∫0π2 (|sin(x)-cos(2x)|)dx Since sin(x)−cos(2x)≤0,[0,π6]sin(x)-cos(2x)≤0,[0,π6] and sin(x)−cos(2x)≥0,[π6,π2]sin(x)-cos(2x)≥0,[π6,π2] So, the integral can be split as,…

1 educator answer

**MATH**

Given the coordinates (2, 0), (0, 2), and (-1, 1). Let A=(2, 0), B(0, 2), and C(-1, 1). Find the equation of line AB using A(2, 0) and B(0, 2). The slope of line AB is 2−00−2=−12-00-2=-1 The…

1 educator answer

**MATH**

Vertices of the triangle are (0,0) , (3,1) and (1,2) Equation of the line through (0,0) and (3,1) is, y−0=(1−03−0)(x−0)y-0=(1-03-0)(x-0) y=x3y=x3Equation of the line through (0,0) and(1,2) is,…

1 educator answer

**MATH**

**y=(14)x2,y=2×2,x+y=3,x>0y=(14)x2,y=2×2,x+y=3,x>0 Sketch the region enclosed by the given curves and…**

Given y=14×2,y=2×2,x+y=3,x>0y=14×2,y=2×2,x+y=3,x>0 Find the intersection point of y=2x2y=2×2 and y=−x+3.y=-x+3. 2×2=−x+32×2=-x+3 2×2+x−3=02×2+x-3=0 (2x+3)(x−1)=0(2x+3)(x-1)=0 x=−32,x=1x=-32,x=1 Ignore the x=-3/2. The original problem…

1 educator answer

**MATH**

**y=1x,y=x,y=(14)x,x>0y=1x,y=x,y=(14)x,x>0 Sketch the region enclosed by the given curves and find…**

Given y=1x,y=x,y=14x,x>0y=1x,y=x,y=14x,x>0 Find the intersection point of y=xy=x and y=1x.y=1x. x=1xx=1x x2=1×2=1 x=1x=1 When x=1, y=1. The intersection point is (1, 1). Find the intersection point of…

1 educator answer

**MATH**

**y=|x|,y=x2−2y=|x|,y=x2-2 Sketch the region enclosed by the given curves and find its area.**

You need to find the intersection points beteen the curves, hence, since |x| = y, you need to discuss two cases, y = -x and y = x. For y = -x, you need to find the intersection between curves by…

1 educator answer

**MATH**

**y=x−−√,y=(12)x,x=9y=x,y=(12)x,x=9 Sketch the region enclosed by the given curves and find its…**

You need to determine first the points of intersection between curves y=x−−√y=x and y=(12)xy=(12)x , by solving the equation, such that: x−−√=(12)x⇒x=x24⇒x24−x=0x=(12)x⇒x=x24⇒x24-x=0…

1 educator answer

**MATH**

**y=cos(x),y=1−cos(x),0≤x≤πy=cos(x),y=1-cos(x),0≤x≤π Sketch the region enclosed by the given curves…**

y=cos(x),y=1−cos(x),0≤x≤πy=cos(x),y=1-cos(x),0≤x≤π Refer the attached image, y=cos(x) is plotted in red color and y=1-cos(x) is plotted in blue color. From the graph, cos(x) is above (1-cos(x)) from 0 to pi/3…

1 educator answer

**MATH**

y=cos(x),y=sin(2x),x=0,x=π2y=cos(x),y=sin(2x),x=0,x=π2 Refer the attached image, y=cos(x) is plotted in red color and y=sin(2x) is plotted in blue color. From graph, cos(x) is above sin(2x) from 0 to pi/6 sin(2x)…

1 educator answer

**MATH**

**y=x3,y=xy=x3,y=x Sketch the region enclosed by the given curves and find its area.**

You need to determine first the points of intersection between curves y=x3y=x3 and y=xy=x , by solving the equation, such that: x3=x⇒x3−x=0x3=x⇒x3-x=0 Factoring out x yields:

1 educator answer

**MATH**

y=tan(x),y=2sin(x),−π3≤x≤π3y=tan(x),y=2sin(x),-π3≤x≤π3 Refer the attached image. Graph of y=tan(x) is plotted in blue color and graph of y=2sin(x) is plotted in red color. Area of the region enclosed by the…

1 educator answer

**MATH**

**x=y4,y=2−x−−−−−√,y=0x=y4,y=2-x,y=0 Sketch the region enclosed by the given curves and find…**

x=y4,y=2−x−−−−−√,y=0x=y4,y=2-x,y=0 Refer the attached image. x=y4x=y4 is plotted in red color and y=2−x−−−−−√y=2-x is plotted in blue color.The curves intersect at x=y=1. y=2−x−−−−−√⇒y2=2−x⇒x=2−y2y=2-x⇒y2=2-x⇒x=2-y2…

1 educator answer

**MATH**

**y=cos(πx),y=4×2−1y=cos(πx),y=4×2-1 Sketch the region enclosed by the given curves and find its area.**

y=cos(πx),y=4×2−1y=cos(πx),y=4×2-1 Refer the attached image. Graph of cos(pix) is plotted in blue color and graph of y=4x^2-1 is plotted in red color. From the graph , the curves intersect at x=±± 1/2. Area…

1 educator answer

**MATH**

**y=x−1−−−−−√,x−y=1y=x-1,x-y=1 Sketch the region enclosed by the given curves and find its area.**

First, you need to find the point of intersection between the curves y=x−1−−−−−√y=x-1 and y=x−1y=x-1 , by solving the equation: x−1−−−−−√=x−1⇒x−1=(x−1)2⇒(x−1)2−(x−1)=0x-1=x-1⇒x-1=(x-1)2⇒(x-1)2-(x-1)=0…

1 educator answer