8. Definite Integrals

Which of the following limits is equal to the definite integral from $2$ to $5$ of $1+x^4$ $dx$?

Find the area of the surface defined by $z=2\surd x^3+y^3$ over the region $0\le x\le 1$, $0\le y\le 1$.

Given the function $f\left(x\right)$ graphed below, where $f\left(x\right)$ is a straight line from $(0,0)$ to $(2,2)$, evaluate the definite integral ${\int }_0^{2}f\left(x\right)dx$.

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

Evaluate the iterated integral: $\int \underset{0}{}\stackrel{2}{}\int \underset{0}{}\stackrel{y^2}{}{x}^{2}ydxdy$

Let $g\left(x\right)={\int }_0^{x}f\left(t\right)dt$, where $f$ is the function whose graph is shown. Which of the following statements is true about $g\left(x\right)$?

Calculate the double integral of $f(x,y)=x+y$ over the region $R$, where $R$ is the rectangle defined by $0\le x\le 2$ and $1\le y\le 3$.

Evaluate the integrals in Exercises 47–68.

∫₁² 4 dv

v²

Evaluate the surface integral ${\int }_S^{}(x^{2}z+y^{2}z) dS$, where $S$ is the hemisphere $x^2+y^2+z^2=4$, $z\ge 0$.

Evaluate the iterated integral: ${\int }_0^4{\int }_0^{y^2}{x}^{2}ydxdy$

Determine the area of the region bounded by $y=x+\sin x$, the x-axis, $x=0$, and $x=\pi$.

Given the parametric equations $x=t^2-3t$ and $y=t$ for $t\in [0,3]$, what is the area enclosed by the curve and the y-axis?

Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.

―∫₄¹ 2ƒ(𝓍) d𝓍

If ∫₀² ƒ(x) dx = π, ∫₀² 7g(x) dx = 7, and ∫₀¹ g(x) dx = 2, find the value of each of the following.

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d. ∫₀² √2ƒ(x) dx

114. {Use of Tech} Arc length of the natural logarithm Consider the curve y = ln(x).

a. Find the length of the curve from x = 1 to x = a and call it L(a).

(Hint: The change of variables u = sqrt(x^2 + 1) allows evaluation by partial fractions.)