8. Definite Integrals

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(a) If ƒ is symmetric about the line 𝓍 = 2 , then ∫₀⁴ ƒ(𝓍) d𝓍 = 2 ∫₀² ƒ(𝓍) d𝓍.

Write the two definite integrals subtracted below as a single integral.

${\int }_1^6\sqrt{x^2-5x}dx-{\int }_{10}^6\sqrt{x^2-5x}dx$

Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.

(a) Evaluate H (0) .

(b) Find the average value of ƒ shown in the figure on the interval [2,6] and then find the point(s) c in (2, 6) guaranteed to exist by the Mean Value Theorem for Integrals. 

Evaluate the line integral of the vector field $F(x,y)=(x+9y,x^2)$ along the curve $C$, where $C$ is the line segment from $(0,0)$ to $(1,2)$:${\int }_C^{}(x+9y)dx+x^{2}dy$

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 $0\le t\le 2\pi$.