# Calculus 1: Applications of Integration

Find the area bounded by the following curves/lines

$y = x + 1$

$y = 9 - x^2$

$x = -1$

$x = 2$

Find the area between the curves $y = x$ and $y = x^2$

Compute the area of the region bounded by the curves $y = x^3$ and $y = 3x - 2$

Find the average value on $[0, 16]$ of $f(x) = \sqrt{x}$

What is the average value of the function $f(x) = 3x^2 - 2x$ on $[1, 4]$

Use the shell method to determine the volume formed by the bounded region rotated about the x-axis.

$y = x^2$, $y = 0$, $x = 2$

Let R be the region enclosed by the graph of $f(x) = x^4 - 2.3x^3 + 4$ and the horizontal line y = 4, as shown in the figure above.

A. Find the volume of the solid generated when R is rotated about the horizontal line y = -2

B. Region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an isosceles right triangle with a leg in R. Find the volume of the solid.

C. The vertical line x = k divides R into two regions with equal areas. Write, but do not solve, an equation involving integral expressions whose solution gives the value k.