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Definite integral as an area: Example 2 Find the value of the integral Z 4 (4 − x) dx. 0 A. 8 B. 4 C. 2 D. 1 Math 105 (Section 204) Integral Calculus – Definite integrals 2011W T2 1/5 Definite integral as an area: Example 3 Find the value of the integral Z 2π sin x dx. 0 A. 1 B. 0 C. 2π D. −2π Math 105 (Section 204) Integral Calculus – Definite integrals 2011W T2 2/5 Approximating areas under curves For an arbitrary function f , how does one compute Math 105 (Section 204) Integral Calculus – Definite integrals Rb a f (x) dx? 2011W T2 3/5 Approximating areas under curves For an arbitrary function f , how does one compute Rb a f (x) dx? Riemann sum Given a function y = f (x) on the interval [a, b], its Riemann sum Rn (f ) is an expression of the form Rn (f ) = n X k=1 f (xk ) b−a , n b−a where xk is some point in the subinterval Ik = [a + (k − 1) b−a n , a + k n ]. Math 105 (Section 204) Integral Calculus – Definite integrals 2011W T2 3/5 Approximating areas under curves For an arbitrary function f , how does one compute Rb a f (x) dx? Riemann sum Given a function y = f (x) on the interval [a, b], its Riemann sum Rn (f ) is an expression of the form Rn (f ) = n X k=1 f (xk ) b−a , n b−a where xk is some point in the subinterval Ik = [a + (k − 1) b−a n , a + k n ]. The value of Rn (f ) depends on the choice of xk within Ik . If xk = left (resply right) endpoint of Ik for all k, then Rn (f ) is called the left (resply right) Riemann sum. Math 105 (Section 204) Integral Calculus – Definite integrals 2011W T2 3/5 What does a Riemann sum represent? Partition [a, b] into n equal subintervals Ik , 1 ≤ k ≤ n. Math 105 (Section 204) Integral Calculus – Definite integrals 2011W T2 4/5 What does a Riemann sum represent? Partition [a, b] into n equal subintervals Ik , 1 ≤ k ≤ n. On each Ik , replace f by a constant function whose value on the entire subinterval is f (xk ). Math 105 (Section 204) Integral Calculus – Definite integrals 2011W T2 4/5 What does a Riemann sum represent? Partition [a, b] into n equal subintervals Ik , 1 ≤ k ≤ n. On each Ik , replace f by a constant function whose value on the entire subinterval is f (xk ). Get a step function, i.e., a piecewise constant curve that approximates f . Math 105 (Section 204) Integral Calculus – Definite integrals 2011W T2 4/5 What does a Riemann sum represent? Partition [a, b] into n equal subintervals Ik , 1 ≤ k ≤ n. On each Ik , replace f by a constant function whose value on the entire subinterval is f (xk ). Get a step function, i.e., a piecewise constant curve that approximates f . The Riemann sum Rn (f ) is the area under this “approximating” curve, and as such an approximation of the area under f (see accompanying diagram). Math 105 (Section 204) Integral Calculus – Definite integrals 2011W T2 4/5 What does a Riemann sum represent? Partition [a, b] into n equal subintervals Ik , 1 ≤ k ≤ n. On each Ik , replace f by a constant function whose value on the entire subinterval is f (xk ). Get a step function, i.e., a piecewise constant curve that approximates f . The Riemann sum Rn (f ) is the area under this “approximating” curve, and as such an approximation of the area under f (see accompanying diagram). Clearly the larger n is, the better the approximation. Math 105 (Section 204) Integral Calculus – Definite integrals 2011W T2 4/5 What does a Riemann sum represent? Partition [a, b] into n equal subintervals Ik , 1 ≤ k ≤ n. On each Ik , replace f by a constant function whose value on the entire subinterval is f (xk ). Get a step function, i.e., a piecewise constant curve that approximates f . The Riemann sum Rn (f ) is the area under this “approximating” curve, and as such an approximation of the area under f (see accompanying diagram). Clearly the larger n is, the better the approximation. Relation between Riemann sums and the definite integral Z b f (x) dx = lim Rn (f ) = lim a Math 105 (Section 204) n→∞ n→∞ n X k=1 Integral Calculus – Definite integrals f (xk ) b−a . n 2011W T2 4/5 An example Which one of the definite integrals below equals the limit lim n→∞ A. Z Z Z Z 1 Math 105 (Section 204) ? 1 x dx 1 + x2 1 1 dx 1 + x2 2 x2 dx 1 + x2 0 D. k2 n2 1 dx 1 + x2 0 C. 1+ 2 1 B. k n2 Integral Calculus – Definite integrals 2011W T2 5/5