How should I solve this calculus question?

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."

Answer 1

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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.