Volume and Area AP Calculus

Front

What is the formula for the area between two curves?

Back

The area A between two curves y=f(x) and y=g(x) from x=a to x=b is given by: A = \int_{a}^{b} (f(x) - g(x)) \, dx.

Front

How do you find the volume of a solid of revolution using the disk method?

Back

The volume V of a solid of revolution generated by rotating a function y=f(x) around the x-axis from x=a to x=b is given by: V = \pi \int_{a}^{b} (f(x))^2 \, dx.

Front

What is the volume of the solid generated by revolving the area bounded by y=x^2 and the x-axis from [0, 2] around the x-axis?

Back

The volume is V = \frac{32\pi}{5}.

Front

What integral would allow you to find the volume of the region bounded by y = 2x^2 and y = 8 around the line y = 11?

Back

The integral is V = \pi \int_{-2}^{2} \left((11 - 2x^2)^2 - 9\right) \, dx.

Front

What is the formula for the area of a region enclosed by two curves?

Back

A = \int_{a}^{b} (f(x) - g(x)) \, dx, where f(x) is the upper curve and g(x) is the lower curve.

Front

What is the volume of the solid generated when R is rotated about the x-axis?

Back

The volume is V = 8\pi.

Front

What is the formula for the volume of a solid of revolution using the washer method?

Back

The volume V is given by: V = \pi \int_{a}^{b} \left(R(x)^2 - r(x)^2\right) \, dx, where R(x) is the outer radius and r(x) is the inner radius.

Front

What is the volume of the solid generated by rotating the region enclosed by y = sin(x) and the x-axis, from x = 0 to x = π about the x-axis?

Back

The volume is V = \frac{1}{2}\pi.

Front

What is the formula for the area of a triangle?

Back

The area A of a triangle is given by: A = \frac{1}{2} \times base \times height.

Front

What is the relationship between the area and the definite integral?

Back

The area under a curve from x=a to x=b can be found using the definite integral: A = \int_{a}^{b} f(x) \, dx.