Explanation The given equation is: 9 x 2 − 4 y 2 = 36 In the above equation diving by 36 on the both sides. x 2 4 − y 2 9 = 1 The above equation shows in the form of hyperbola whose focus lies on the x- axis. Comparing this with the standard for ( x − h ) 2 b 2 − ( y − k ) 2 a 2 = 1 with a center at ( h , k ) . h = 0 , k = 0 , a 2 = 4 , and b 2 = 9 The vertices are, ( h ± a , k ) = ( 0 ± 2 , 0 ) = ( 2 , 0 ) and ( − 2 , 0 ) The distance between the focus and the center is: c = a 2 + b 2 = 4 + 9 = 13 When the triangular area is in the first quadrant between the curve, and this area is rotated about the x- axis, then the volume of the generated solid can be calculated when x is such that 2 ≤ x ≤ 4

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Chapter 11.6, Problem 75E

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The volume of the solid triangular quadrant bounded by the x-axis, the line x=4 and the hyperbola 9x2−4y2=36 is revolved about the x-axis.

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Chapter 11.6, Problem 74E

Chapter 11.6, Problem 76E

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1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps (each step must be justified). Theorem 0.1 (Abel's Theorem). If y1 and y2 are solutions of the differential equation y" + p(t) y′ + q(t) y = 0, where p and q are continuous on an open interval, then the Wronskian is given by W (¥1, v2)(t) = c exp(− [p(t) dt), where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or W (y1, y2)(t) = 0 for every t in I. 1. (a) From the two equations (which follow from the hypotheses), show that y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0, 2. (b) Observe that Hence, conclude that (YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0. W'(y1, y2)(t) = yY2 - Y1 y2- W' + p(t) W = 0. 3. (c) Use the result from the previous step to complete the proof of the theorem.

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Question 4 Find an equation of (a) The plane through the point (2, 0, 1) and perpendicular to the line x = y=2-t, z=3+4t. 3t, (b) The plane through the point (3, −2, 8) and parallel to the plane z = x+y. (c) The plane that contains the line x = 1+t, y = 2 − t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1. (d) The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1+t, and z = 2-t. (e) The plane that contains the lines L₁: x = 1 + t, y = 1 − t, z = 2t and L2 : x = 2 − s, y = s, z = 2.

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The volume of the solid triangular quadrant bounded by the x- axis, the line x = 4 and the hyperbola 9 x 2 − 4 y 2 = 36 is revolved about the x- axis.

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The volume of the solid triangular quadrant bounded by the x-axis, the line x=4 and the hyperbola 9x2−4y2=36 is revolved about the x-axis.

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