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

# 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…

• 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:

• 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.

• 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:

• 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:

• 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…

• MATH

y=sin2(x),y=sin2(x), y=0,y=0, 0≤x≤π.0≤x≤π. Use a computer algebra system to find the exact volume of…

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…

• 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…

• 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…

• 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:

• 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…

• 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…

• 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:

• 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

• 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…

• 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:

• 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:

• 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:

• 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…

• 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…

• 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:

• 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:

• 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:

• 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.

• 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:

• 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:

• 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:

• MATH

y=cos(x),y=x+2sin4(x)y=cos(x),y=x+2sin4(x) Graph the region between the curves and use your calculator…

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…

• 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…

• 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…

• 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….

• 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…

• 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…

• 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…

• 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…

• 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.

• MATH

∫π20(|sin(x)−cos(2x)|)dx∫0π2(|sin(x)-cos(2x)|)dx Evaluate the integral and interpret it as the area of a…

∫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,…

• MATH

(2,0),(0,2),(−1,1)(2,0),(0,2),(-1,1) Use calculus to find the area of the triangle with the given vertices.

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…

• MATH

(0,0),(3,1),(1,2)(0,0),(3,1),(1,2) Use calculus to find the area of the triangle with the given vertices.

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,…

• 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…

• 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…

• 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…

• 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…

• 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…

• MATH

y=cos(x),y=sin(2x),x=0,x=π2y=cos(x),y=sin(2x),x=0,x=π2 Sketch the region enclosed by the given curves…

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)…

• 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:

• MATH

y=tan(x),y=2sin(x),(−π3)≤x≤(π3)y=tan(x),y=2sin(x),(-π3)≤x≤(π3) Sketch the region enclosed by the given…

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…

• 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…

• 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…