The region R is bounded by the curves and

MATHEMATICS · Single correct · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

The region R is bounded by the curves y=x2y = x^2 and y=4x−x2y = 4x - x^2. Two students set up definite integrals to find its area. Student 1: ∫02[(4x−x2)−x2] dx\displaystyle\int_{0}^{2} \bigl[(4x - x^2) - x^2\bigr]\,dx Student 2: ∫04[y−(2−y)] dy\displaystyle\int_{0}^{4} \bigl[\sqrt{y} - (2 - \sqrt{y})\bigr]\,dy Which of the following statements best justifies why both integrals must produce the same numerical value? (a) Because dy=dxdy = dx for differentiable functions, merely changing the variable name cannot alter the value of a definite integral. (b) They compute the signed area between the same two curves, and the area of a region in the plane is independent of the variable of integration. (c) Changing the order of integration in a single-variable integral never affects its value provided the limits are adjusted accordingly. (d) The region between the curves is symmetric about x=2x = 2, so integrating with respect to xx or yy must give the same result.

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