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."

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  • 1 month ago
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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.

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