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Evaluating a Definite Integral In Exercises \(55-62\) , evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{0}^{1} x^{3}\left(2 x^{4}+1\right)^{2} d x $$

Find the average value of the function over the given interval and all values of \(x\) in the interval for which the function equals its average value. \(f(x)=\cos x, \quad\left[0, \frac{\pi}{2}\right]\)

Approximation In Exercises \(53-56\) , determine which value best approximates the definite integral. Make your selection on the basis of a sketch. $$ \begin{array}{l}{\int_{0}^{9}(1+\sqrt{x}) d x} \\ {\begin{array}{llll}{\text { (a) }-3} & {\text { (b) } 9} & {\text { (c) } 27} & {\text { (d) } 3}\end{array}}\end{array} $$

Finding Area by the Limit Definition In Exercises \(55-60,\) use the limit process to find the area of the region bounded by the graph of the function and the \(y\) -axis over the given \(y\) -interval. Sketch the region. $$ f(y)=y^{2}, \quad 0 \leq y \leq 5 $$

Vertical Motion In Exercises \(56-58,\) use \(a(t)=-9.8\) meters per second per second as the acceleration due to gravity. (Neglect air resistance.) A baseball is thrown upward from a height of 2 meters with an initial velocity of 10 meters per second. Determine its maximum height.

Determining Integrability Determine whether the function $$ f(x)=\frac{1}{x-4} $$ is integrable on the interval \([3,5] .\) Explain.

Evaluating a Definite Integral In Exercises \(55-62\) , evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{1}^{2} 2 x^{2} \sqrt{x^{3}+1} d x $$

Vertical Motion In Exercises \(56-58,\) use \(a(t)=-9.8\) meters per second per second as the acceleration due to gravity. (Neglect air resistance.) The Grand Canyon is 1800 meters deep at its deepest point. A rock is dropped from the rim above this point. Write the height of the rock as a function of the time in seconds. How long will it take the rock to hit the canyon floor?

Finding a Function Give an example of a function that is integrable on the interval \([-1,1],\) but not continuous on \([-1,1] .\)

Finding Area by the Limit Definition In Exercises \(55-60,\) use the limit process to find the area of the region bounded by the graph of the function and the \(y\) -axis over the given \(y\) -interval. Sketch the region. $$ f(y)=4 y-y^{2}, \quad 1 \leq y \leq 2 $$