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MATH 1A, 8-9am section Quiz 10, April 20 Name: Student ID: You have 20 min to complete this quiz. Please state your answers clearly and precisely. 1. Given that 03 f (x)dx = 7 and 13 f (x)dx = 3, find 01 f (x)dx (3 points) R R R Solution: We know that 03 f (x)dx = 01 f (x)dx + 13 f (x)dx, so R1 R3 R3 0 f (x)dx = 0 f (x)dx − 1 f (x)dx = 7 − 3 = 4 R R R R √ 2. Evaluate the integral 02 4 − x2 dx by interpreting it as an area (3 points) √ Solution: The function f (x) = 4 − x2 for 0 ≤ x ≤ 2 defines a quarter of the circle with radius 2 and center at the origin. So if we interprete the intergral as the area under the curve, the value R2 √ will be 0 4 − x2 dx = 14 π22 = π 3. Using the definition of the integral, evalute 01 (1 + x2 )dx. You may P want to use that ni=1 i2 = n(n+1)(2n+1) (4 points) 6 2 R1 P 1 2 n Solution: 0 (1 + x )dx = limn→∞ i=1 1 + ni n = R h P i 1 1 Pn n 2 1 + ( ) i i=1 i=1 n3 i h n limn→∞ 1 + 61 (1 + n1 )(2 + n1 ) = 1 limn→∞ = limn→∞ 1 + + 1 3 1 n(n+1)(2n+1) n3 6 =