10. Area and Average Value

Homework

In each of the following problems, use the Fundamental Theorem of Calculus to find the area under the graph of \(f\), above the \(x\)-axis, between \(x=a\) and \(x=b\).

1. \(f(x)=x^3+3x\),   \(a=1\) and \(b=3\).

2. \(f(x)=\dfrac{1}{x}\),   \(a=2\) and \(b=4\). Simplify.

3. \(f(x)=xe^{-x^2}\),   \(a=0\) and \(b=2\).

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4. Sketch the region between the graphs of \(f(x)=-x^2\) and \(g(x)=-3x\). Then find the area between them.

5. Sketch the region bounded by the curves \(y=5-x^2\) and \(y=(x-1)^2\). Then find the area between them.

6. Sketch the region bounded by the curves \(y=x^3-9x\) and \(y=16x-x^3\). Then find the area between them.

7. Sketch the region(s) between the graph of \(x=4y^3\) and the line \(x=y\). Then compute the area between the curves.

8. Find the average value of the function \(f(x)=x^2-4\) on the interval \([-2,6]\). Plot the function and its average.

9. Find the value(s) of \(c\) guaranteed by the Mean Value Theorem for Integrals for the function \(g(x)=\dfrac{1}{x^2}\) on the interval \([2,4]\). Plot the function and the value of \(c\).