How should I solve this calculus question?
I can't figure out how to solve this question. Even just a start would be really helpful.
"Find the area of the region S in the first quadrant bounded by the function y=xsinx, the line y=-2x+5, and the y-axis."
- az_lenderLv 71 month agoFavorite Answer
Roughly speaking, the area is triangular (though one boundary is curved).
The three vertices are
(A) the intersection of y = x*sin(x) with the y-axis, which is at (0,0).
(B) the intersection of y = -2x + 5 with the y-axis, which is at (0,5).
(C) the intersection of y = -2x + 5 with y = x*sin(x), which is near x = 1.66937. More on that down below.
Anyway, so you want to integrate
[(-2x + 5) - x*sin(x)] dx
from x = 0 to x = 1.66937.
The indefinite integral is -x^2 + 5x + x*cos(x) - sin(x).
Can use a calculator to finish.
How did I find x = 1.66937? I found it with a graphing calculator! But you COULD try it by setting
-2x + 5 = x*sin(x) =>
5 = x*[sin(x) + 2].
Because sin(x) is between 0 and 1, the quantity in the brackets is between 2 and 3, so the "x" must be between 5/3 and 5/2. That would give you a hint to start with something like x = 2, and use Newton's method, or some other way of improving your estimate.