Front

What is the formula for the area between two curves y=f(x) and y=g(x) from x=a to x=b?

Back

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

Front

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

Back

The volume V is given by V = \pi \int_{a}^{b} [f(x)]^2 \, dx, where f(x) is the function being revolved around the x-axis.

Front

What is the washer method for finding volumes of solids of revolution?

Back

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

Front

What is the formula for the volume of a solid generated by revolving a region around the y-axis?

Back

The volume V is given by V = \pi \int_{c}^{d} [f(y)]^2 \, dy, where f(y) is the function expressed in terms of y.

Front

How do you determine the limits of integration when finding the area between curves?

Back

The limits of integration are found by solving the equation f(x) = g(x) to find the points of intersection.

Front

What is the significance of the first quadrant in area and volume problems?

Back

The first quadrant is where both x and y are non-negative, simplifying calculations for area and volume.

Front

What is the relationship between the area under a curve and the volume of a solid of revolution?

Back

The area under a curve can be used to calculate the volume of a solid of revolution by integrating the area of circular disks or washers.

Front

How do you set up an integral for the volume of a solid formed by revolving a region around the x-axis?

Back

Use the disk method: V = \pi \int_{a}^{b} [f(x)]^2 \, dx, where f(x) is the height of the region.

Front

What is the formula for the volume of a cylinder?

Back

The volume V of a cylinder is given by V = \pi r^2 h, where r is the radius and h is the height.

Front

What is the formula for the volume of a cone?

Back

The volume V of a cone is given by V = \frac{1}{3} \pi r^2 h, where r is the radius of the base and h is the height.