Applications of Integration — Quiz 1
Questions in this quiz (10)
Calculate the area bounded by $$f(x)=x^4-4x^2$$ and the x-axis. What important conclusion can be drawn from this problem?
- Area equals total signed integral
- Area requires splitting at zeros
- Area is always positive without splitting
- Area equals derivative of the function
Calculate the area bounded by $$y=x^3-2x$$ and $$y=(x+1)^2$$ in the second quadrant.
- $$2$$
- $$\frac52$$
- $$\frac74$$
- $$\frac{11}{12}-\frac{2\sqrt2}{3}$$
Calculate the area in the first quadrant bounded by $$x=4$$ and $$y=x^2+1$$ twice—once with horizontal rectangles and once with vertical ones.
- $$\frac{76}{3}$$
- $$12$$
- $$\frac{32}{3}$$
- $$16$$
Calculate the area bounded by $$x=y^2+2y-3$$ and $$y=-x+2$$.
- $$\frac92$$
- $$6$$
- $$\frac{29\sqrt{29}}{6}$$
- $$5$$
Calculate the area bounded by $$y=2x^3-6x,\ y=5x^2,\ x=0,$$ and $$x=-1$$.
- $$\frac73$$
- $$\frac56$$
- $$\frac52$$
- $$\frac83$$
Find the volume generated by revolving the area bounded by the curves $$y=\sqrt{x+2},\ x=3,$$ and $$y=1$$ about the x-axis.
- $$2\pi$$
- $$3\pi$$
- $$4\pi$$
- $$8\pi$$
Find the volume generated by revolving the area in the first quadrant bounded by $$y=x^2$$ and $$x=2$$ about the line $$x=2$$.
- $$\frac{8\pi}{3}$$
- $$\frac{5\pi}{3}$$
- $$4\pi$$
- $$\frac{32\pi}{3}$$
Find the volume generated by revolving the region bounded by $$y=(x+1)^3,\ y=0,$$ and $$x=0$$ about $$x=2$$.
- $$\frac{15\pi}{4}$$
- $$6\pi$$
- $$\frac{11\pi}{10}$$
- $$5\pi$$
Find the volume generated by revolving the region bounded by $$y=\sin x$$, $$y=0$$, and $$x=\frac{\pi}{3}$$ about the x-axis. Then find $$c$$ such that the volume is split in half.
- $$c=\frac{\pi}{6}$$
- $$c=\frac{\pi}{4}$$
- $$c=\frac{\pi}{5}$$
- $$c=\frac{\pi}{3}$$
What volume results if you rotate the region in the first quadrant bounded by $$y=x^2+2$$, $$x=1$$, and $$y=2$$ about the line $$y=1$$?
- $$\frac{5\pi}{3}$$
- $$2\pi$$
- $$\frac{7\pi}{3}$$
- $$\frac{13\pi}{15}$$