A car travels at a constant velocity of $12\text{ m/s}$ for seconds, and then at $20\text{ m/s}$ for $3\le t\le 7$ seconds. Graph the velocity function and determine the total displacement of the car from $t=0$ to $t=7$ seconds.

Suppose that for real numbers $p_1$, $p_2$, ..., $p_{15}$ we have $\sum _{k=1}^{15}{p}_k=30$. What is the value of $\sum _{k=1}^{15}\frac{p_k}{6}$?

The graph of $h\left(x\right)$ is given below. Compute ${\int }_0^{3}h\left(x\right)dx$ using geometry.

Evaluate ${\int }_0^4(3x+2)dx$ using right Riemann sums and the definition of the definite integral.

Write the left and right Riemann sums in sigma notation for an arbitrary value of $n$ to approximate the definite integral ${\int }_2^4\sqrt{1+x^2}dx$.

Apply Simpson’s Rule to approximate ${\int }_0^{\frac{\pi }{2}}{e}^{-2x}\cos xdx$ with $n=8$ subintervals.

Evaluate ${\int }_1^{3}4x^{2}dx$ and ${\int }_{-3}^{3}4x^{2}dx$.

Let $B\left(x\right)={\int }_b^{x}f\left(t\right)dt$, and $f\left(t\right)=4t+1$. Assuming that $f$ and $f^{\prime }$ are continuous functions for all real numbers, is $B\left(x\right)$ a quadratic function?

Determine the arc length of the curve $y=\ln \left(\cos x\right)$ from $x=0$ to $x=\frac{\pi }{4}$.

Compute ${\int }_1^e\frac{7}{y}dy$ using the Fundamental Theorem of Calculus, Part 2.

Find $\frac{d}{dx}{\int }_{-1}^x(3t^2-2t)dt$.

Find $\frac{d}{ds}{\int }_{s^4}^5\sqrt{u^3+4}du$.