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Chapter 5: Problem 3

Find the area between the graph of \(f\) and the \(x\) -axis. $$(x)=(x+2)^{-2}, \quad x \in[0,2]$$

Short Answer

Expert verified
The area between the curve \(f(x) = (x + 2)^{-2}\) and the x-axis from 0 to 2 is \(\frac{1}{4}\) square units.

Step by step solution

01

Set Up the Integral

The integral representing the area between this curve \(f(x) = (x + 2)^{-2}\) and x-axis from 0 to 2 would be set up as follows: \[\int_{0}^{2} (x+2)^{-2} dx.\]
02

Perform the Integration

The antiderivative of \((x+2)^{-2}\) is \(-1/(x+2)\). Thus the definite integral becomes: \[-\left. \frac{1}{x+2} \right\vert_{0}^{2}.\]
03

Evaluate the Definite Integral

Plug in the upper limit first, then subtract the value obtained by plugging in the lower limit: \[-\left. \frac{1}{x+2} \right\vert_{0}^{2} = -\frac{1}{4} - (-\frac{1}{2}) = \frac{1}{4}.\]

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