Set up an integral for the volume of the solid obtained by rotating

Question

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the speci-ted line. Then use your calculator to evaluate the integral cor rect to hive decimal places.

31. y = e^-x^2 , y = 0, x = - 1, x = 1

(a) About the x-axis

(b) about y=-1

33.x^2 +4y^2 =4

(a) About y = 2

(b) x = 2

53. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 3 cm, 4 cm, and 5 cm

Answer 1

31. y = e^(-x^2), y = 0, x = -1, x = 1:

(a) About the x-axis:

To find the volume of the solid generated by rotating the region bounded by the given curves about the x-axis, we can use the disk method. The volume V can be found using the disk method:

V = pi * ∫(from -1 to 1) (e^(-x^2))^2 dx

V = pi * ∫(from -1 to 1) e^(-2x^2) dx

Use a calculator to evaluate the integral:

V ≈ pi * 0.88208 = 2.76894 * pi

(b) About y = -1:

To find the volume of the solid generated by rotating the region bounded by the given curves about the line y = -1, we can use the washer method. The volume V can be found using the washer method:

V = pi * ∫(from -1 to 1) [(1 + e^(-x^2))^2 - 1^2] dx

V = pi * ∫(from -1 to 1) (1 + 2e^(-x^2) + e^(-2x^2) - 1) dx

V = pi * ∫(from -1 to 1) (2e^(-x^2) + e^(-2x^2)) dx

Use a calculator to evaluate the integral:

V ≈ pi * 1.76416 = 5.53787 * pi

33. x^2 + 4y^2 = 4:

(a) About y = 2:

First, we need to solve for x in terms of y:

x^2 = 4 - 4y^2

x = ±sqrt(4 - 4y^2)

Since the region of interest is symmetric about the y-axis, we can consider only the positive square root and multiply the result by 2:

x = sqrt(4 - 4y^2)

Now, we need to find the limits of integration. The ellipse x^2 + 4y^2 = 4 intersects y = 2 when:

x^2 + 4(2)^2 = 4

x^2 = -12 (no solution)

Thus, the limits of integration are y = -1/2 to y = 1/2. The volume V can be found using the washer method:

V = 2 * pi * ∫(from -1/2 to 1/2) (sqrt(4 - 4y^2))^2 dy

V = 2 * pi * ∫(from -1/2 to 1/2) (4 - 4y^2) dy

Use a calculator to evaluate the integral:

V ≈ 2 * pi * 2 = 4 * pi

(b) x = 2:

The region of interest is symmetric about the x-axis, so the volume is 0.

53. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 3 cm, 4 cm, and 5 cm:

The volume of a tetrahedron with mutually perpendicular edges can be found using the formula:

V = (1/3) * base_area * height

In this case, the edges are the base and the height, so the formula becomes:

V = (1/3) * 3 * 4 * 5

V = 20 cm³