Calculus

Approximation of Integrals

1. Using a right-hand Riemann sum with 4 subdivisions, estimate $\displaystyle\int_0^2(2x-x^2)dx$.

2. Use Simpson's Rule with $n=6$ to estimate $\displaystyle\int_0^2x^3dx$.

3. Use the Trapezoidal rule with 4 subintervals and the table below to estimate $\displaystyle\int_1^9h(x)dx$.
4. \begin{array}{|c|c|c|c|c|c|} \hline x & 1 & 3 & 5 & 7 & 9 \\ \hline h(x) & 2 & 3 & 3 & 4 & 5\\ \hline \end{array}

5. Use a midpoint Riemann sum with $n=4$ to estimate $\displaystyle\int_0^2(x^2+2)dx$.

Integrals with the FTC

1. $\displaystyle\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\sec^2xdx=$

2. $\displaystyle\int_0^{\frac{\pi}{2}}\cos x \sin(\sin x)dx=$ (hint: your answer will have cosine in it)

3. $\displaystyle\int_3^5(3-|x-2|)dx=$

4. $\displaystyle\int_2^{e+1}\left(\frac{4}{x-1}\right)dx=$

5. $\displaystyle\int_{\frac{1}{\pi}}^{\frac{e^4}{\pi}}\frac{\ln(\pi x)}{x}dx=$

6. Find the positive difference between $\displaystyle\frac{d}{dx}\left[\displaystyle\int_0^3(2x+5)dx\right]$ and $\displaystyle\int_0^3\left[\frac{d}{dx}(2x+5)\right]dx$.

Fun Integrals

1. Evaluate $\displaystyle\lim_{n \to \infty}\sum_{i=1}^n\left[2\left(1+\frac{3i}{n}\right)+1\right]\frac{3}{n}$.

2. If $F(x)=\displaystyle\int_0^{3x}e^{t^4}dt$, find $F'(0)$.

3. $\displaystyle\frac{d}{dx}\int_{2x}^{5x}\cos{t}dt=$

4. If $F(x)=\displaystyle\int_{x^4+e^x}^{(x+1)^2+\sin{x}}(t^2-2)dt$, compute $F'(0)$.

5. $\displaystyle\int_0^{\infty}xe^{-x^2}dx=$

6. If $\displaystyle\int_0^3f(x)dx=4$, $\displaystyle\int_3^6f(x)dx=4$, and $\displaystyle\int_2^6f(x)dx=5$, find $\displaystyle\int_0^2f(x)dx$.

7. If $\displaystyle\int_{-2}^4f(x)dx=a$ and $\displaystyle\int_3^4f(x)dx=b$, what is $\displaystyle\int_3^{-2}f(x)dx$ equal to (in terms of $a$ and $b$)?