Question

Let R and S in the figure above be defined as follows: R is the region in the first and second quadrants bounded by the graphs of y=3-x^2 and y=2^x. S is the shaded region in the quadrant bounded by the two graphs, the x-axis and the y-axis.

Answer 1

a. Find the area of S.
b. Find the volume of the solid generated when R is rotated about the horizontal line y= -1
c. The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an isosceles right triangle with one leg across the base of the solid. Write, but do not evaluate, an integral expression that gives the volume of the solid.

Answer 2

equate the curves to find the points of intersection

2^x = 3 - x^2 the point of intersection in the 1st quadrant it x = 1

the curve y = 3 - x^2 has zeros at \[x = \pm \sqrt{3}\]

to the area S is \[A = \int\limits_{0}^{1} 2^x dx + \int\limits_{1}^{\sqrt{3}} 3 - x^2 dx\]

Answer 3

hello