Mathematics

Problem

Let

C_{1}

be the curve

y = x^{3} - 4x

and let

C_{2}

be the line

y = x

. The two curves intersect in three distinct points, enclosing one finite region. a. (i) Find the coordinates of the three points of intersection of

C_{1}

and

C_{2}

. b. (ii) Show that the finite region bounded only by the two curves lies entirely between the leftmost and rightmost intersection points found in part (i). c. (iii) Taking into account any change in which curve is above the other, set up all the definite integral(s) required to find the exact area of this bounded region. d. (iv) Evaluate the integral(s) obtained in part (iii) to determine the exact area.

Intuitive explanation

The key is recognizing that when two curves intersect at multiple points, the finite region between them is bounded by those intersection points. Find where $C_1$ and $C_2$ meet by setting the equations equal, then use the intersection points as your integration limits. The area is the integral of the absolute difference between the curves—whichever is on top minus the one below—over that interval.

Intuitive explanation

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