Overview
1) Use the limit definition of the definite integral with right Riemann sums to evaluate the following definite integral.
[Image unavailable: LaTeX: \int^1_0\left(4x-2\right)dx]
2) Suppose [Image unavailable: LaTeX: \int^4_1f\left(x\right)dx=6,\:\:\int^4_1g\left(x\right)dx=4,\:\:and\:\int^4_3f\left(x\right)dx=2]. Evaluate the following integrals.
a) [Image unavailable: LaTeX: \int^4_1\left(3f\left(x\right)-2g\left(x\right)\right)dx]
b) [Image unavailable: LaTeX: \int^3_1f\left(x\right)dx]
3) Evaluate the following derivative. [Image unavailable: LaTeX: \frac{d}{dx}\int^{e^{2x}}_3\cos\left(t^2\right)dt\:]
4) Evaluate the following integrals.
a) [Image unavailable: LaTeX: \int^2_{-2}\left(3x^4-2x+1\right)dx]
b) [Image unavailable: LaTeX: \int^{\frac{\pi}{3}}_{\frac{\pi}{6}}\left(\sec^2t+\csc^2t\right)dt\:]
c) [Image unavailable: LaTeX: \int\cos\left(3x\right)dx]
d) [Image unavailable: LaTeX: \int^5_{-5}\frac{w^3}{\sqrt[]{w^{50}+w^{20}+1}}dw\:\:\left[Hint.\:\:Use\:Symmetry\right]]
e) [Image unavailable: LaTeX: \int^1_0\frac{v^3+1}{\sqrt[]{v^4+4v+4}}dv]
f) [Image unavailable: LaTeX: \int e^{\sin^2x}\sin2xdx\:\:\left[Hint.\:\:\sin2x=2\sin xcosx\right]\:]
5) Find the average value of the following function on the given interval. [Image unavailable: LaTeX: f\left(x\right)=\frac{1}{x}\:on\:\left[1,e\right]]