Problem 52
Graphing Calculator Challenge If \(k > 1 ,\) and if the average value of \(x ^ { k }\) on \([ 0 , k ]\) is \(k ,\) what is \(k ?\) Check your result with a CAS if you have one available.
Problem 53
In Exercises \(49-54,\) use NINT to solve the problem. For what value of \(x\) does \(\int_{0}^{x} e^{-t^{2}} d t=0.6 ?\)
Problem 53
Show that if \(F ^ { \prime } ( x ) = G ^ { \prime } ( x )\) on \([ a , b ] ,\) then \(F ( b ) - F ( a ) = G ( b ) - G ( a )\)
Problem 53
In Exercises \(47-56,\) use graphs, your knowledge of area, and the fact that \(\quad\) $$\int_{0}^{1} x^{3} d x=\frac{1}{4}$$ to evaluate the integral. $$\int_{0}^{2}\left(\frac{x}{2}\right)^{3} d x$$
Problem 54
In Exercises \(47-56,\) use graphs, your knowledge of area, and the fact that \(\quad\) $$\int_{0}^{1} x^{3} d x=\frac{1}{4}$$ to evaluate the integral. $$\int_{-8}^{8} x^{3} d x$$
Problem 55
In Exercises \(47-56,\) use graphs, your knowledge of area, and the fact that \(\quad\) $$\int_{0}^{1} x^{3} d x=\frac{1}{4}$$ to evaluate the integral. $$\int_{0}^{1}\left(x^{3}-1\right) d x$$
Problem 55
In Exercises 55 and \(56,\) find \(K\) so that $$\int_{a}^{x} f(t) d t+K=\int_{b}^{x} f(t) d t$$ $$f(x)=x^{2}-3 x+1 ; \quad a=-1 ; \quad b=2$$
Problem 56
In Exercises 55 and \(56,\) find \(K\) so that $$\int_{a}^{x} f(t) d t+K=\int_{b}^{x} f(t) d t$$ $$f(x)=\sin ^{2} x ; \quad a=0 ; \quad b=2$$
Problem 56
In Exercises \(47-56,\) use graphs, your knowledge of area, and the fact that \(\quad\) $$\int_{0}^{1} x^{3} d x=\frac{1}{4}$$ to evaluate the integral. $$\int_{0}^{1} \sqrt[3]{x} d x$$
Problem 57
The function
\(f(x)=\left\\{\begin{array}{ll}{\frac{1}{x^{2}},} & {0
