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Area Under a Curve for Definite Integrals for AP Calculus

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By — McGraw-Hill Professional
Updated on Oct 24, 2011

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:

Area Under a Curve

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:

Area Under a Curve

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.

Area Under a Curve

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 ) = x2 – 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.

Area Under a Curve

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.

Area Under a Curve

Example 4

Using a calculator, find the area bounded by f(x ) = x3 + x2 – 6x and the x-axis. See Figure 12.3-6.

Area Under a Curve

Step 1.  Enter y1 =x ^3 + x ^2 – 6x.

Step 2.  Enter

Example 5

The area under the curve y = ex from x = 0 to x = k is 1. Find the value of k.

Area =

ln(ek) = 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 ≤ xk is twice the area of the region kx ≤ π, find k. (See Figure 12.3-7.)

Area Under a Curve

Area Under a Curve

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

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