AP Calculus AB/BC - Unit 8: Applications of Integration

NOTE: $(a, b)$ are x-coordinates and $(c, d)$ are y-coordinates.

Slices ⊥ to x-axis:

$$A = \int_a^b [f(x) - g(x)] \, dx$$

Slices ⊥ to y-axis:

$$A = \int_c^d [f(y) - g(y)] \, dy$$

About x-axis:

$$V = \pi \int_a^b [R(x)]^2 \, dx$$

About y-axis:

$$V = \pi \int_c^d [R(y)]^2 \, dy$$

About x-axis:

$$V = \pi \int_a^b \left( [R(x)]^2 - [r(x)]^2 \right) dx$$

About y-axis:

$$V = \pi \int_c^d \left( [R(y)]^2 - [r(y)]^2 \right) dy$$

Where **base** is the distance between the two curves and $a$ and $b$ are the limits of integration.

SQUARES:

$$V = \int_a^b (\text{base})^2 \, dx$$

TRIANGLES:

$$V = \frac{\sqrt{3}}{4} \int_a^b (\text{base})^2 \, dx$$

ISOSCELES RIGHT TRIANGLES:

$$V = \frac{1}{4} \int_a^b (\text{base})^2 \, dx$$

RECTANGLES:

$$V = \int_a^b (\text{base}) \cdot h \, dx$$

where $h$ is the height of the rectangles.

SEMI-CIRCLES:

$$V = \frac{\pi}{2} \int_a^b (\text{radius})^2 \, dx$$

where **radius** is ½ of the distance between the two curves.