8. Definite Integrals

Evaluate the line integral of the function $f(x,y)=xy$ along the curve $C$, where $C$ is given by the parametric equations $x=t^2$, $y=2t$ for $0\le t\le 5$. That is, compute ${\int }_C^{}xyds$.

Evaluate the iterated integral: ${\int }_0^2{\int }_0^1{\left(x+y\right)}^{2}dxdy$

Evaluate the definite integral: ${\int }_1^4(t^3-t^2-4) dt$.

Find the area of the part of the plane $5x+4y+z=20$ that lies in the first octant.

Consider the double integral ${\int }_0^2{\int }_{y^2}^{4}f(x,y)dxdy$. Which of the following correctly expresses the integral with the order of integration reversed?

Suppose the graph of $f$ consists of two regions between $x=0$ and $x=4$: from $x=0$ to $x=2$, $f\left(x\right)$ forms a triangle above the $x$-axis with area $3$; from $x=2$ to $x=4$, $f\left(x\right)$ forms a rectangle below the $x$-axis with area $4$. What is the value of the definite integral ${\int }_0^{4}f\left(x\right)dx$?

Evaluate the definite integral: ${\int }_{-2}^3(x^2-3) dx$.

Evaluate the line integral ${\int }_C^{}(2x+3y) dx+(x-y) dy$, where C is the line segment from $(0,0)$ to $(2,4)$.

Calculate the value of the double integral ${\int }_0^4{\int }_0^1(4x+1+xy) dy dx$.

Given the graph of $f\left(x\right)$ below, evaluate the definite integral ${\int }_0^{4}f\left(x\right)dx$.

What is the average (mean) value of the function $3t^3-t^2$ over the interval $1\le t\le 2$?

Evaluate the definite integral: ${\int }_0^3(1+7x) dx$.

Evaluate the line integral ${\int }_C^{}{y}^{3}ds$, where $C$ is the curve given by $x=t^3$, $y=t$, for $0\le t\le 4$.

Find the exact length of the curve given by $x=2t^3$, $y=t^2-2$ for $0\le t\le 5$.

Which of the following integrals represents the area of the region bounded by the curves $y=x^2$ and $y=4$ between $x=0$ and $x=2$?