91Ó°ÊÓ

In Exercises \(43-46,\) compute \(F^{\prime}\) and \(F^{\prime \prime} .\) Determine the intervals on which \(F\) is increasing, decreasing, concave up, and concave down. $$ F(x)=\int_{0}^{x} t(t-1) d t $$

Calculate \(A=\int_{-1}^{1}|x| d x\). Let \(N\) be an even positive integer that is not divisible by \(4 .\) Show that if a uniform partition of order \(N\) is used, then the Midpoint Rule approximation of \(A\) is exact, but the Simpson's Rule approximation is not.

Evaluate the definite integrals. $$ \int_{1}^{2} \frac{24}{x^{2}} \csc \left(\frac{\pi}{3} \frac{2 x-1}{x}\right) d x $$

Evaluate the given definite integral by finding an antiderivative of the integrand and applying Theorem \(3 .\) $$ \int_{1}^{2} \frac{x^{2}+2 x+1}{x} d x $$

In Exercises \(43-46,\) compute \(F^{\prime}\) and \(F^{\prime \prime} .\) Determine the intervals on which \(F\) is increasing, decreasing, concave up, and concave down. $$ F(x)=\int_{0}^{x} t \exp (-t) d t $$

Find the area of the region(s) between the two curves over the given range of \(x\). $$ f(x)=\left(x^{3}-8\right) / x \quad g(x)=7(x-2), 1 \leq x \leq 4 $$

Calculate the first and second derivatives of \(F(x)=\int_{a}^{u(x)} f(t) d t\) for the given functions \(u\) and \(f\) \(u(x)=\log _{2}(x) \quad f(t)=1 / t\)

In Exercises \(43-46,\) compute \(F^{\prime}\) and \(F^{\prime \prime} .\) Determine the intervals on which \(F\) is increasing, decreasing, concave up, and concave down. $$ F(x)=\int_{1}^{x} t \ln (t) d t \quad 0

Calculate \(\mathcal{R}\left(f, \mathcal{L}_{N}\right),\) the Riemann sum \(\mathcal{R}(f, \mathcal{S})\) using the midpoint of each subinterval for the choice of points, and \(\mathcal{R}\left(f, \mathcal{U}_{N}\right)\). (You will notice that the inequalities of line \((5.2 .2)\) hold. \()\) $$ f(x)=e^{x} \quad I=[0,1], N=1 $$

Suppose that \(p\) and \(q\) are constants with \(p>0\) and \(q>0\) Is it ever true that $$ \int_{0}^{1} x^{p} \cdot x^{q} d x=\int_{0}^{1} x^{p} d x \cdot \int_{0}^{1} x^{q} d x ? $$