Integration & Accumulation of Change

The interval $[0,4]$ is divided into four subintervals of equal length. Which expression gives the right Riemann sum approximation for $\displaystyle\int_{0}^{4}f(x)\,dx$?

• (A) $f(0)+f(1)+f(2)+f(3)$
• (B) $f(1)+f(2)+f(3)+f(4)$
• (C) $f(0.5)+f(1.5)+f(2.5)+f(3.5)$
• (D) $\tfrac{1}{2}\bigl[f(0)+2f(1)+2f(2)+2f(3)+f(4)\bigr]$

If $\displaystyle\int_{0}^{5}f(x)\,dx=12$ and $\displaystyle\int_{0}^{2}f(x)\,dx=3$, then $\displaystyle\int_{2}^{5}f(x)\,dx=$

• (A) $-9$
• (B) $9$
• (C) $15$
• (D) $36$

If $F(x)=\displaystyle\int_{1}^{x}t^{2}\,dt$, then $F'(x)=$

• (A) $\dfrac{x^{3}}{3}$
• (B) $x^{2}-1$
• (C) $x^{2}$
• (D) $2x$

If $F(x)=\displaystyle\int_{0}^{x^{2}}\sin t\,dt$, then $F'(x)=$

• (A) $\sin(x^{2})$
• (B) $2x\sin(x^{2})$
• (C) $-\cos(x^{2})+1$
• (D) $\cos(x^{2})\cdot 2x$

$\displaystyle\int (3x^{2}-4x+5)\,dx =$

• (A) $x^{3}-2x^{2}+5x+C$
• (B) $x^{3}-2x^{2}+5+C$
• (C) $6x-4+C$
• (D) $\dfrac{x^{3}}{3}-\dfrac{4x^{2}}{2}+5x+C$

$\displaystyle\int 2x\,(x^{2}+1)^{4}\,dx =$

• (A) $\dfrac{(x^{2}+1)^{5}}{5}+C$
• (B) $\dfrac{(x^{2}+1)^{5}}{10}+C$
• (C) $2x\cdot\dfrac{(x^{2}+1)^{5}}{5}+C$
• (D) $\dfrac{x^{2}(x^{2}+1)^{5}}{5}+C$

$\displaystyle\int_{0}^{\pi/2}\cos x\,dx =$

• (A) $0$
• (B) $\dfrac{1}{2}$
• (C) $1$
• (D) $\dfrac{\pi}{2}$

The graph of $f$ consists of two line segments (forming a triangle above the $x$-axis with vertices $(-2,0)$, $(0,2)$, $(2,0)$) and a semicircle of radius $2$ below the $x$-axis on $[2,6]$. Find $\displaystyle\int_{-2}^{6}f(x)\,dx$.

• (A) $4-2\pi$
• (B) $4+2\pi$
• (C) $3+2\pi$
• (D) $3-2\pi$

$\displaystyle\int_{0}^{1}\dfrac{x}{(x^{2}+1)^{2}}\,dx =$

• (A) $\dfrac{1}{4}$
• (B) $\dfrac{1}{2}$
• (C) $\dfrac{1}{8}$
• (D) $\dfrac{1}{6}$

Which definite integral equals $\displaystyle\lim_{n\to\infty}\sum_{i=1}^{n}\!\left(1+\dfrac{2i}{n}\right)^{2}\!\cdot\dfrac{2}{n}$?

• (A) $\displaystyle\int_{0}^{2}x^{2}\,dx$
• (B) $\displaystyle\int_{1}^{3}x^{2}\,dx$
• (C) $\displaystyle\int_{1}^{2}(1+x)^{2}\,dx$
• (D) $\displaystyle\int_{0}^{1}(1+2x)^{2}\,dx$

$f$ is positive, increasing, and concave down on $[a,b]$. Which of the following must be an underestimate of $\displaystyle\int_{a}^{b}f(x)\,dx$?

• (A) Left Riemann sum
• (B) Right Riemann sum
• (C) Trapezoidal sum
• (D) Midpoint Riemann sum

$\displaystyle\int \dfrac{x^{2}+1}{x+1}\,dx =$

• (A) $\dfrac{x^{2}}{2}-x+2\ln|x+1|+C$
• (B) $x-\ln|x+1|+C$
• (C) $\dfrac{x^{3}+x}{x^{2}+x}+C$
• (D) $\dfrac{x^{2}}{2}+\ln|x+1|+C$

Selected values of the differentiable function $f$ are given. Let $g(x)=\displaystyle\int_{0}^{x}f(t)\,dt$.

Using a midpoint Riemann sum with two subintervals of equal length, the approximation for $g(8)$ is

• (A) $-4$
• (B) $-2$
• (C) $4$
• (D) $12$

Let $g(x)=\displaystyle\int_{0}^{x}f(t)\,dt$, where $f$ is continuous and is positive on $(0,3)$, zero at $x=3$, and negative on $(3,5)$. At what value of $x$ does $g$ attain its maximum on $[0,5]$?

• (A) $0$
• (B) $3$
• (C) $5$
• (D) Cannot be determined

Water flows into a tank at a rate $r(t)$ gallons per minute, where $t$ is in minutes. The most appropriate units of $\displaystyle\int_{0}^{10}r(t)\,dt$ are

• (A) gallons per minute
• (B) gallons per minute squared
• (C) gallons
• (D) minutes per gallon

$\displaystyle\int \dfrac{1}{x^{2}+2x+5}\,dx =$

• (A) $\dfrac{1}{2}\arctan\!\left(\dfrac{x+1}{2}\right)+C$
• (B) $\arctan(x+1)+C$
• (C) $\ln|x^{2}+2x+5|+C$
• (D) $\dfrac{1}{4}\ln|x^{2}+2x+5|+C$

Which technique is most appropriate for evaluating $\displaystyle\int x\sec^{2}(x^{2})\,dx$?

• (A) Power rule (direct antidifferentiation)
• (B) $u$-substitution with $u=x^{2}$
• (C) Long division
• (D) Completing the square

If $\displaystyle\int_{1}^{4}\bigl[\,2f(x)-3\,\bigr]\,dx = 14$ and $\displaystyle\int_{1}^{4}f(x)\,dx = ?$

• (A) $\dfrac{17}{2}$
• (B) $\dfrac{23}{2}$
• (C) $\dfrac{11}{2}$
• (D) $7$