Definite Integrals — Quiz 2

Questions in this quiz (10)

1. Consider the graph of $$y=6x-x^2$$. Using a midpoint Riemann sum with $$3$$ equal width subintervals, find the approximate area of the shaded region above.

   • $$19$$
   • $$36$$
   • $$38$$
   • $$54$$

2. Let $$A=\int_0^1\cos x\,dx$$. We estimate $$A$$ using the $$L$$, $$R$$, and $$T$$ approximations with $$n=100$$ subintervals. Which is true?

   • $$L$$ < $$A$$ < $$T$$ < $$R$$
   • $$L$$ < $$T$$ < $$A$$ < $$R$$
   • $$R$$ < $$A$$ < $$T$$ < $$L$$
   • $$R$$ < $$T$$ < $$A$$ < $$L$$

3. The average value of $$\sec^2x$$ over the interval from $$x=\frac{\pi}{6}$$ to $$x=\frac{\pi}{3}$$ is:

   • $$\frac{3}{\pi}$$
   • $$\frac{4\sqrt{3}}{\pi}$$
   • $$\frac{6}{\pi}(\sqrt{3}-1)$$
   • $$3(\sqrt{3}-1)$$

4. $$\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}\sin^2x\cos xdx$$ is equal to:

   • $$\frac{1}{3}$$
   • $$\frac{7}{24}$$
   • $$\frac{1}{2}$$
   • $$\frac{2}{3}$$

5. Choose the Riemann sum whose limit is the integral:$$\int_0^2\cos(4x)dx$$

   • $$\lim_{n\to\infty}\sum_{k=1}^n\cos\left(\frac{4k}{n}\right)\left(\frac{1}{n}\right)$$
   • $$\lim_{n\to\infty}\sum_{k=1}^n\cos\left(\frac{8k}{n}\right)\left(\frac{2}{n}\right)$$
   • $$\lim_{n\to\infty}\sum_{k=1}^n\cos\left(\frac{4k}{n}\right)\left(\frac{2}{n}\right)$$
   • $$\lim_{n\to\infty}\sum_{k=1}^n\cos\left(\frac{8k}{n}\right)\left(\frac{1}{n}\right)$$

6. Find the average value of function $$f$$, as shown in the graph above, on the interval $$[0,5]$$.

   • $$6$$
   • $$7$$
   • $$8$$
   • $$5$$

7. The integral $$\int_{-3}^{3}\sqrt{9-x^2}dx$$ gives the area of:

   • a circle of radius $$3$$
   • a semicircle of radius $$3$$
   • a quadrant of a circle of radius $$3$$
   • half of an ellipse

8. If $$f(x)$$ is continuous on the closed interval $$[p,q]$$, then there exists at least one number $$c$$, $$p<c<q$$, such that $$\int_p^qf(x)dx$$ is equal to:

   • $$\frac{f(c)}{q-p}$$
   • $$f(c)(q-p)$$
   • $$\frac{f'(c)}{q-p}$$
   • $$f'(c)(q-p)$$

9. If $$F(u)=\int_2^u(3-x^2)^4dx$$ then $$F'(u)$$ is equal to:

   • $$-2u(3-u^2)^3$$
   • $$(3-u^2)^4-1$$
   • $$(3-u^2)^4$$
   • $$-u(3-u^2)^4$$

10. Let $$g(x)=\int_2^{4x}f(t)dt$$. Then $$g'(3)=$$?

    • $$4f(12)$$
    • $$f(12)$$
    • $$3f(12)$$
    • $$12f(12)$$