Mathematics

Mathematics

Problem

Region RR is bounded by the curves x=(y−1)2x=(y-1)^{2} and x=y+1x=y+1 and by the horizontal lines y=−1y=-1 and y=4y=4. Which of the following integral expressions equals the area of RR? a) ∫−14[(y+1)−(y−1)2] dy\displaystyle \int_{-1}^{4}[(y+1)-(y-1)^{2}]\,dy b) ∫−10[(y−1)2−(y+1)] dy+∫03[(y+1)−(y−1)2] dy+∫34[(y−1)2−(y+1)] dy\displaystyle \int_{-1}^{0}[(y-1)^{2}-(y+1)]\,dy+\int_{0}^{3}[(y+1)-(y-1)^{2}]\,dy+\int_{3}^{4}[(y-1)^{2}-(y+1)]\,dy c) ∫−10[(y−1)2−(y+1)] dy+∫04[(y+1)−(y−1)2] dy\displaystyle \int_{-1}^{0}[(y-1)^{2}-(y+1)]\,dy+\int_{0}^{4}[(y+1)-(y-1)^{2}]\,dy d) ∫−11[(y−1)2−(y+1)] dy+∫14[(y+1)−(y−1)2] dy\displaystyle \int_{-1}^{1}[(y-1)^{2}-(y+1)]\,dy+\int_{1}^{4}[(y+1)-(y-1)^{2}]\,dy

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