# Calculus _ Larson Ch 4.3a - 4.3a RIEMANN SUMS AND DEFINITE...

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4.3a RIEMANN SUMS AND DEFINITE INTEGRALS Name: Area Under a Curve as a Definite Integral If is nonnegative and integrable over a closed interval [ a , b ], then the area under the curve y = f ( x ) over [ a , b ] is the integral of f from a to b , A = ( ) f x dx a b # . In Exercises 1–4, express the area of the region in the figure as definite integral. 1. 2. 3. 4. In Exercises 5–8, sketch the region whose area is given by the definite integral. Then evaluate the integral by regarding it as the area under the graph of a function. 5. dx 6 1 5 - # 6. ( ) x dx 2 6 3 2 + - # 7. x dx 1 0 3 - # 8. ( ) x dx 3 4 2 2 2 + - - # - 1 1 - 2 3 1 3 x y 2 4 5 2 5 4 y = 3 - 1 1 - 2 3 1 3 x y 2 4 5 2 5 4 3 x - 4 y = - 11 - 1 1 - 2 3 1 3 x y 2 4 5 2 5 4 ( x - 1) 2 + ( y - 1) 2 = 9 - 2 2 - 4 6 1 3 x y 4 8 10 2 5 4 x = y 3 1. _________________ 2. _________________ 3. _________________ 4. _________________ 5. _________________ 6. _________________ 7. _________________ 8. _________________
9. The graph of g consists of two straight lines and a semicircle, as shown in the figure right. Evaluate each integral by interpreting it in terms of areas. a. ( ) g x dx 0 2 # b. ( ) g x dx 2 6 # c. ( ) g x dx 0 7 # Any sum of the form ( ) f x x k n 1 Δ = / is called a Riemann sum 10. Evaluate the Riemann sum for f ( x ) = 2 x - x 2 , 0 x k .