91Ó°ÊÓ

Let \(f(x)=9-x^{2}\) for \(-2 \leq x \leq 3,\) and let \(P\) be the regular partition of [-2,3] into five subintervals. Find the Riemann sum \(R_{p}\) if \(f\) is evaluated at the midpoint of each subinterval of \(P\).

A point \(P\) is moving on a coordinate line with a continuous acceleration function \(a\). If \(v\) is the velocity function, then the average acceleration on a time interval \(\left[t_{1}, t_{2}\right]\) is $$ \frac{v\left(t_{2}\right)-v\left(t_{1}\right)}{t_{2}-t_{1}} $$ Show that the average acceleration is equal to the average value of \(a\) on \(\left[t_{1}, t_{2}\right]\).

Evaluate the integral if \(a\) and \(b\) are constants. $$ \int\left(\frac{a}{b^{2}} t\right) d t $$

Exer. 9-48: Evaluate the integral. $$ \int \frac{x}{\cos ^{2}\left(x^{2}\right)} d x $$

Evaluate the integral if \(a\) and \(b\) are constants. $$ \int(a+b) d u $$

Exer. 9-48: Evaluate the integral. $$ \int x \cot \left(x^{2}\right) \csc \left(x^{2}\right) d x $$

Verify the inequality without evaluating the integrals. $$ \int_{0}^{1} x^{2} d x \geq \int_{0}^{1} x^{3} d x $$

Verify the inequality without evaluating the integrals. $$ \int_{1}^{2} x^{2} d x \leq \int_{1}^{2} x^{3} d x $$

Evaluate the integral if \(a\) and \(b\) are constants. $$ \int\left(b-a^{2}\right) d u $$

Exer. 9-48: Evaluate the integral. $$ \int \sec \left(\frac{x}{3}\right) \tan \left(\frac{x}{3}\right) d x $$