Riemann Sums & Definite Integrals (College Board AP® Calculus AB): Exam Questions

The function $f$ is continuous on the closed interval $\left[0.9\right]$ and has values that are given in the table above. Using a trapezoidal sum with three subintervals and the values from the table, what is the approximation of ${\int }_0^{9}f\left(x\right)dx$?

If $F\left(x\right)={\int }_0^x\sqrt{t^3+9}dt$, then $F^{\text{'}}\left(3\right)=$

What are all the values of $k$ for which ${\int }_4^k{x}^{2}dx=0$?

If ${\int }_3^{12}f\left(x\right)dx=7$ and ${\int }_6^{3}f\left(x\right)dx=2$, then ${\int }_6^{12}f\left(x\right)dx=$

If a trapezoidal sum underapproximates ${\int }_0^{4}f\left(x\right)dx$, and a left Riemann sum overapproximates ${\int }_0^{4}f\left(x\right)dx$, which of the following could be the graph of $y=f\left(x\right)$?

If $f\left(x\right)=mx+b$ and $c<x<d$ then ${\int }_c^d{f}^{\text{'}\text{'}}\left(x\right)dx=$

Let $F\left(x\right)$ be an antiderivative of $\sin x{\left(1+\cos x\right)}^2$. If $F\left(1\right)=2$, then $F\left(3\right)=$

For $-\frac{\pi }{2}<x<\frac{\pi }{2}$, if $f\left(x\right)={\int }_x^0{\sec }^{2}tdt$ then $f\text{'}\left(x\right)=$

The graph of a piece-wise linear function $f$ is shown above for $-2\le x\le 3$. What is the value of ${\int }_{-2}^{3}f\left(x\right)dx$?

Which of the following is equivalent to the definite integral ${\int }_{-2}^3{x}^{2}dx$?

The graph of a differentiable function $f$ is shown above. If $h\left(x\right)={\int }_0^{x}f\left(t\right)dt$, which of the following is true?

The figure above shows the graph of $f$. If $f\left(x\right)={\int }_3^{x}g\left(t\right)dt$, which of the following could be the graph of $y=g\left(x\right)$?

If $f^{\text{'}}\left(x\right)<0$ for all real numbers $x$ and ${\int }_3^{6}f\left(t\right)dt=0$, which of the following could be a table of values for the function $f$?

Let $f$ be a continuous function. If $G\left(x\right)$ is an antiderivative for $f\left(x\right)$ and $G\left(7\right)=-3$, then $G\left(1\right)=$

$k$is a constant. If ${\int }_0^k\left(2k{x}^3+5x^4\right)dx=48$, then $k=$

${\int }_0^5\left|x-4\right|dx=$

The graph of the function $f$ is shown above for $0\le x\le 4$. Which of the following has the greatest value?

$\frac{d}{dx}\left({\int }_0^{x^3}\frac{1}{t^4+1}dt\right)=$

Let $f$ be the function given by $f\left(x\right)={\int }_0^x\cos \left(t^2\right)dt$ for $-1\le x\le 1.5$. On which of the following intervals is $f$ increasing?