8.1 Areas and Volumes

Question 1

what is the formula to calculate the area between two curves?

Answer 1

A=∫(a on the bottom and b on the top)[f(x)−g(x)]dx, where f(x) is the upper curve and g(x) is the lower curve.

Question 2

What should you do if the curves intersect within the interval?

Answer 2

Split the interval at the points of intersection, and calculate the area for each region separately, ensuring you subtract the lower function from the upper in each region.

Question 3

What is the Disk Method for finding volumes of solids of revolution?

Answer 3

Rotate a curve around an axis and use the formula V=π∫(a on the bottom, b on the top) [f(x)]^2 dx for rotation about the x-axis.

Question 4

How do you apply the Washer Method for volumes of solids with a hole?

Answer 4

Use the formula 𝑉=([R(x)]^2
−[r(x)]^2)dx, where R(x) is the outer radius and r(x) is the inner radius.

Question 5

What is the Shell Method?

Answer 5

A method for calculating the volume of a solid of revolution by integrating cylindrical shells

Question 6

When should you use the Washer Method?

Answer 6

When the solid of revolution has a hole (like a donut shape)

Question 7

When is the Shell Method typically used?

Answer 7

When rotating around the y-axis or when the Disk/Washer Method would be more complex (e.g., functions of
𝑥).

Question 8

How do you determine whether to use the Disk/Washer or Shell Method?

Answer 8

Use the Disk/Washer Method when slicing perpendicular to the axis of rotation, and use the Shell Method when slicing parallel to the axis of rotation.

Question 9

What is the formula for the volume using the Shell Method when rotating about the y-axis?

Answer 9

V=2π∫(a on the bottom, b on the top) [x⋅f(x)]dx, where f(x) is the height of the shell and x is the distance from the
y-axis.

Question 10

What should you do when calculating the area between curves that are functions of y?

Answer 10

Use the formula
A=∫(c on the bottom, d on the top) [f(y)−g(y)]dy, where f(y) is the right curve and g(y) is the left curve.

Question 11

What is the key difference between the Disk and Washer Methods?

Answer 11

The Disk Method assumes the solid has no hole, while the Washer Method accounts for an inner hole by subtracting the volume of the inner function from the outer function.