Section II Part A — Full Practice Exam 2

Questions in this quiz (2)

1. Let $$R$$ be the shaded region in the first quadrant enclosed by the $$y$$-axis and the graphs of $$y=1-x^3$$ and $$y=\sin\left(x^2\right),$$ as shown in the figure above.$$(a)$$ Find the area of $$R.$$$$(b)$$ A horizontal line, $$y=k,$$ is drawn through the point where the graphs of $$y=1-x^3$$ and $$y=\sin\left(x^2\right)$$ intersect. Find $$k$$ and determine whether this line divides $$R$$ into two regions of equal area. Show the work that leads to your conclusion.$$(c)$$ Find the volume of the solid generated when $$R$$ is revolved about the line $$y=-3.$$

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2. A planetary rover travels on a flat surface. The path of the rover for the time interval $$0\le t\le2$$ hours is shown in the rectangular coordinate system above. The rover starts at the point with coordinates $$\left(6,5\right)$$ at time $$t=0$$. The coordinates $$\left(x\left(t\right),y\left(t\right)\right)$$ of the position of the rover change at rates given by$$x'\left(t\right)=-12\sin\left(2t^2\right)$$$$y'\left(t\right)=10\cos\left(1+\sqrt{t}\right),$$where $$x\left(t\right)$$ and $$y\left(t\right)$$ are measured in meters and $$t$$ is measured in hours.$$(a)$$ Find the acceleration vector of the rover at time $$t=1$$. Find the speed of the rover at time $$t=1.$$$$(b)$$ Find the total distance that the rover travels over the time interval $$0\le t\le1.$$$$(c)$$ Find the $$y$$-coordinate of the position of the rover at time $$t=1.$$$$(d)$$ The rover receives a signal at each point where the line tangent to its path has slope $$\frac{1}{2}$$. At what times $$t,$$ for $$0\le t\le2,$$ does the rover receive a signal?

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