8. Definite Integrals

Find the exact length of the curve $y=\frac{2}{3}(1+x^2){(}^{\frac{3}{2}}$ for $0\le x\le 4$.

Given the curve $y=x^2$ from $x=0$ to $x=2$, find the exact length of the curve.

Evaluate the definite integral: ${\int }_0^{\frac{\pi }{6}}\sin \left(t\right){\cos }^2\left(t\right) dt$.

Evaluate the definite integral: $2{\int }_{\frac{\pi }{3}}^{\frac{\pi }{3}}{csc}^2(\frac{1}{2}t)dt$.

Evaluate the line integral ${\int }_C^{}\frac{x}{y} ds$, where C is the curve given by $x=t^3$, $y=t^4$, for $1\le t\le 2$.

Evaluate the double integral ${\int }_D^{}(2x+y)da$, where $D=\left\{\right(x,y)\mid 1\le y\le 4,y-3\le x\le 3\}$.

Let C be the curve parameterized by $x=t^2$, $y=t^3$ for $0\le t\le 1$. Find the value of the line integral ${\int }_C^{}ydx$.

Evaluate the definite integral: ${\int }_1^5{w}^2\ln \left(w\right) dw$

Evaluate the definite integral: ${\int }_0^1(6x^2-6x+8) dx$.

Find the exact length of the curve given by $x=5\cos \left(t\right)-\cos \left(5t\right)$, $y=5\sin \left(t\right)-\sin \left(5t\right)$, for $0\le t\le 2\pi$.

Evaluate the definite integral: ${\int }_1^2{w}^2\ln \left(w\right) dw$.

Express the following limit as a definite integral on the interval $[0,10]$.

${\lim }_{n\to \infty }{\sum }_{k=1}^n{(x_k^{\ast }-3)}^2\Delta x$

Evaluate the double integral ${\int }_0^2{\int }_0^1(2x+1+xy) dy dx$.

Evaluate the double integral ${\int }_0^2{\int }_0^3(4-x-y)dydx$ by first identifying it as the volume of a solid. What is the value of the integral?

Given the parametric equations $x={\sin }^2\left(t\right)$, $y=8\cos \left(t\right)$ for $0\le t\le \pi$, what is the area enclosed by the curve and the y-axis?