a Use calculus to find the circumference of a circle with radius r b Use

# A use calculus to find the circumference of a circle

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(a) Use calculus to find the circumference of a circle with radius r . (b) Use calculus to sum the areas between concentric circles and find the area of a circle. Hint: Show dA = 2 πx dx (c) Use polar coordinates and double integrals to find the area of a circle. 3-45. (a) Find common area of intersection associated with the circles x 2 + y 2 = r 2 0 and ( x - r 0 ) 2 + y 2 = r 2 0 (b) Find the volume of the solid of revolution if this area is rotated about the x - axis. Hint: Make use of symmetry. 3-46. Sketch the region R over which the integration is to be performed ( a ) 1 0 y y 2 f ( x, y ) dx dy ( b ) π 0 3 cos θ 0 f ( r, θ ) r dr dθ ( c ) 4 1 x 2 1 f ( x, y ) dy dx 3-47. Sketch the region of integration, change the order of integration and evaluate the integral. ( a ) 3 1 2 y - 1 12 xy dx dy ( b ) 4 0 x x/ 2 3 xy dy dx ( c ) b a d c xy dx dy, a x b c y d 3-48. Integrate the function f ( x, y ) = 3 2 xy over the region R bounded by the curves y = x and y 2 = 4 x (a) Sketch the region of integration. (b) Integrate with respect to x first and y second. (c) Integrate with respect to y first and x second.
268 3-49. Evaluate the double integral and sketch the region of integration. I = 1 0 x 0 2( x + y ) dy dx. 3-50. For f ( x ) = x and b > a > 0 , find the number c such that the mean value theorem b a f ( x ) dx = f ( c )( b - a ) is satisfied. Illustrate with a sketch the geometrical interpretation of your result. 3-51. Make appropriate substitutions and show dx α 2 + x 2 = 1 α tan - 1 x α + C dx α 2 - x 2 = 1 α tanh - 1 x α + C, x < α dx b 2 + ( x + a ) 2 = 1 b tan - 1 x + a b + C dx b 2 - ( x + a ) 2 = 1 b tanh - 1 x + a b + C, x + a < b 3-52. (a) If f ( x ) = f ( x + T ) for all values of x , show that nT 0 f ( x ) dx = n T 0 f ( x ) dx (b) If f ( x ) = - f ( T - x ) for all values of x , show that T a f ( x ) dx = - a 0 f ( x ) dx 3-53. (a) Use integration by parts to show e ax sin bx dx = 1 a e ax sin bx - b a e ax cos bx dx and e ax cos bx dx = 1 a e ax cos bx + b a e ax sin bx dx (b) Show that e ax sin bx dx = e ax a sin bx - b cos bx a 2 + b 2 + C e ax cos bx dx = e ax b sin bx + a cos bx a 2 + b 2 + C 3-54. Make an appropriate substitution to evaluate the given integrals ( a ) ( e 2 x + 3) m e 2 x dx ( b ) e 4 x + e 3 x e x + e - x dx ( c ) ( e x + 1) 2 e x dx 3-55. Evaluate the given integrals ( a ) x + a x 3 dx ( b ) ( x + a )( x + b ) x 3 dx ( c ) ( x + a )( x + b )( x + c ) x 3 dx 3-56. Evaluate the given integrals ( a ) ln(1 + x ) dx ( b ) x 4 + 1 x - 1 dx ( c ) ( a + bx )( x + c ) m dx
269 3-57. Use a limiting process to evaluate the given integrals ( a ) 0 e - st dt, s > 0 ( b ) x -∞ e t dt ( c ) 0 te - st dt, s > 0 3-58. Consider a function J n ( x ) defined by an infinite series of terms and having the representation J n ( x ) = x n 2 n 1 n ! - x 2 2 2 1!( n + 1)! + x 4 2 4 2!( n + 2)! + · · · + ( - 1) m x 2 m 2 2 m m !( n + m )! + · · · where n is a fixed integer and m represents the m th term of the series. Here m takes on the values m = 0 , 1 , 2 , . . . . Show that x 0 J 1 ( x ) dx = 1 - J 0 ( x ) The function J n ( x ) is called the Bessel function of the first kind of order n . 3-59. Determine a general integration formula for I n = x n e x dx and then evaluate the integral I 4 = x 4 e x dx 3-60.

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