Explanation Given: The area A of the region in the first quadrant bounded by the line y = 1 2 x , the x- axis and the ellipse 1 9 x 2 + y 2 = 1 is equal to the area of the region bounded by the line y = m x , the y- axis and the the ellipse 1 9 x 2 + y 2 = 1 in the first quadrant. Formula used: Integral of ∫ a 2 − x 2 d x = 1 2 x a 2 − x 2 + a 2 2 sin − 1 x a and integral of ∫ x n d x = 1 n + 1 x n + 1 . Calculation: Since, A is the area of the region in the first quadrant bounded by the line y = 1 2 x , the x- axis and the ellipse 1 9 x 2 + y 2 = 1 . Substitute the equation y = 1 2 x in the equation of ellipse to find the point of intersection. That is, 1 9 x 2 + ( x 2 ) 2 = 1 4 x 2 + 9 x 2 36 = 1 13 x 2 = 36 x = 6 13 Draw a perpendicular from the point of intersection to the x- axis, that intersects x- axis at the point ( 6 13 , 0 ) . Therefore, the area A is: A = ∫ 0 6 13 x 2 d x + ∫ 6 13 3 9 − x 2 3 d x = 1 2 ∫ 0 6 13 x d x + 1 3 ∫ 6 13 3 3 2 − x 2 d x = 1 2 [ x 2 2 ] 0 6 13 + 1 3 [ 1 2 x 3 2 − x 2 + 3 2 2 sin − 1 x 3 ] 6 13 3 = 1 4 [ 36 13 ] + 1 6 [ 9 ( π 2 ) − ( 6 13 × 9 13 + 9 sin − 1 2 13 ) ] Further solve and get, A = 9 13 + 1 6 [ 9 π 2 − 54 13 − 9 sin − 1 2 13 ] = 9 13 + 3 π 4 − 9 13 − 3 2 sin − 1 2 13 = 3 π 4 − 3 2 sin − 1 2 13 Now, let the area of the region bounded by the line y = m x , the y- axis and the the ellipse 1 9 x 2 + y 2 = 1 in the first quadrant is denoted by B. So, the area B of the region in the first quadrant bounded by the line y = m x , the y- axis and the ellipse 1 9 x 2 + y 2 = 1 . Substitute the equation x = y m , in the equation of ellipse to find the point of intersection

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Author: Larson, Ron, Edwards, Bruce

Publisher: Cengage Learning,

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Chapter 14.8, Problem 41E

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To calculate: The positive value of m for which the area A of the region bounded by the line y=12x, the x- axis and the ellipse 19x2+y2=1 in the first quadrant is equal to the area of the region bounded by the line y=mx, the y- axis and the the ellipse 19x2+y2=1 in the first quadrant.

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Chapter 14.8, Problem 40E

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The positive value of m for which the area A of the region bounded by the line y = 1 2 x , the x- axis and the ellipse 1 9 x 2 + y 2 = 1 in the first quadrant is equal to the area of the region bounded by the line y = m x , the y- axis and the the ellipse 1 9 x 2 + y 2 = 1 in the first quadrant.

Solution Summary: The author calculates the positive value of m for which the area A of the region in the first quadrant bounded by the lines y=12x, x-axis

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To calculate: The positive value of m for which the area A of the region bounded by the line y=12x, the x- axis and the ellipse 19x2+y2=1 in the first quadrant is equal to the area of the region bounded by the line y=mx, the y- axis and the the ellipse 19x2+y2=1 in the first quadrant.

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