Definite Integrals — Quiz 3

Questions in this quiz (5)

1. A continuous function $$f$$ takes on the values shown in the table above. Estimate $$\int_0^{20}f(x)dx$$ using a left rectangular approximation with four subintervals.

   • $$116$$
   • $$124$$
   • $$108$$
   • $$146$$

2. The average value of$$f(x)=x^2$$, $$x$$ < $$3$$$$f(x)=5x$$, $$x\ge3$$on the interval $$0\le x\le6$$ is:

   • $$12.5$$
   • $$12.75$$
   • $$13.25$$
   • $$14.5$$

3. If $$x=3\cos\theta$$ and $$y=2\sin\theta$$, then $$\int_1^3xy\,dx$$ is equivalent to:

   • $$18\int_{\pi/3}^0\sin\theta\cos^2\theta\,d\theta$$
   • $$18\int_0^{\pi/3}\sin^2\theta\cos\theta\,d\theta$$
   • $$18\int_0^{\pi/3}\sin\theta\cos^2\theta\,d\theta$$
   • $$18\int_0^{\cos^{-1}(1/3)}\sin^2\theta\cos\theta\,d\theta$$

4. $$\int_{-2}^{4}|x|dx$$?

   • $$6$$
   • $$8$$
   • $$10$$
   • $$12$$

5. The area of the shaded region in the figure above is equal to $$\ln3$$. If we approximate $$\ln3$$ using both a left Riemann sum and a right Riemann sum with $$2$$ equal width subintervals as shown, which inequality follows?

   • $$\frac{1}{2}$$ < $$\int_1^2\frac{1}{x}dx$$ < $$1$$
   • $$\frac{1}{3}$$ < $$\int_1^3\frac{1}{x}dx$$ < $$2$$
   • $$\frac{1}{3}$$ < $$\int_2^3\frac{1}{x}dx$$ < $$\frac{1}{2}$$
   • $$\frac{5}{6}$$ < $$\int_1^3\frac{1}{x}dx$$ < $$\frac{3}{2}$$