Practice problems for these concepts can be found at: Areas and Volumes Practice Problems for AP Calculus

If *y* = *f* (*x* ) is continuous and non-negative on [*a*, *b*], then the area under the curve of *f* from *a* to *b* is:

If f is continuous and *f* < 0 on [*a*, *b*], then the area under the curve from *a* to *b* is:

If *x* = *g* (*y* ) is continuous and non-negative on [*c*, *d*], then the area under the curve of *g* from *c* to d is:

### Example 1

Find the area under the curve of *f* (*x* ) = (*x* – 1)3 from *x* = 0 to *x* = 2.

Step 1. Sketch the graph of *f *(*x* ). See Figure 12.3-3.

Step 2. Set up integrals.

Step 3. Evaluate the integrals.

### Example 2

Find the area of the region bounded by the graph of *f *(*x* ) = *x*^{2} – 1, the lines *x* = – 2 and *x* = 2 and the *x*-axis.

Step 1. Sketch the graph of *f*(*x* ). See Figure 12.3-4.

Step 2. Set up integrals.

Step 3. Evaluate the integrals.

### Example 3

Find the area of the region bounded by *x* = *y* 2, *y* = – 1, and *y* = 3. See Figure 12.3-5.

### Example 4

Using a calculator, find the area bounded by *f*(*x* ) = *x*^{3} + *x*^{2} – 6*x* and the *x*-axis. See Figure 12.3-6.

Step 1. Enter *y*1 =*x* ^3 + *x* ^2 – 6*x*.

Step 2. Enter

### Example 5

The area under the curve *y* = *e ^{x}* from

*x*= 0 to

*x*=

*k*is 1. Find the value of

*k*.

Area =

ln(*e ^{k}*) = ln 2;

*k*= ln 2.

### Example 6

The region bounded by the *x* -axis, and the graph of *y* = sin *x* between *x* = 0 and *x* = π is divided into 2 regions by the line *x* = *k*. If the area of the region for 0 ≤ *x* ≤ *k* is twice the area of the region *k* ≤ *x* ≤ π, find *k*. (See Figure 12.3-7.)

Practice problems for these concepts can be found at: Areas and Volumes Practice Problems for AP Calculus

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