Calculus 1: Definite Integrals and Fundamental Theorem

Evaluate the definite integral below

$\displaystyle\int_{-2}^2 \ {x^2 \cos{(\frac{x^3}{8})}} \ dx$

Evaluate the following definite integral

$\displaystyle\int_0^4 \ x \sqrt{x^2 + 9} \ dx$

Find the area under the curve over the interval $[0,4]$

$f(x) = x^2 + 1$

Find the area under the curve over the interval $[1,4]$

$f(x) = \frac{2}{x}$

Evaluate the integral

$\displaystyle\int_0^2 (2x - 2x^2) \ dx$

Prove the fundamental theorem of calculus

Evaluate the integral $\displaystyle\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (2 - \csc^2{x}) \ dx$

Find the area the region bounded by:

$y = 1 + \sqrt[3]{x}$

$x = 0$

$x = 8$

$y = 0$

Compute the definite integrals

$\displaystyle\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \tan (x) \ dx$ and $\displaystyle\int_{\frac{-\pi}{3}}^{\frac{\pi}{3}} \tan (x) \ dx$