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1180 Guided Notes | 5.2: The Definite Integral limn→∞∑i=1nf(xi∗)Δx=limn→∞[f(x1∗)Δx+f(x2∗)Δx+⋯+f(xn∗)Δx]xi∗ is the value I'm using to find the height of rectangle i. This limit becomes very important, and comes up in many different places. We give it a special name and definition: Definition of a Definite Integral: Let f be defined on the closed interval [a,b]. We divide [a,b] into n subintervals of equal width Δx= n b−a . Let x 0 ,x 1 ,x 2 ,…,x n be endpoints of the subintervals where x 0 =a and x n =b. We let x 1 ∗ ,x 2 ∗ ,…,x n ∗ be any sample points in these subintervals. For example, x i ∗ lies in the sub interval i. Then, the Definite Integral of f from a to b is usually, we'll use the right end point of interval i. ∫ a b f(x)dx=lim n→∞ ∑ i=1 n f(xi∗)Δx Provided the limit exists. If it does, we say f is integrable on [a,b]. You will be spending a lot of time with the integral. The integral symbol (a stretched out S for "sum"): ∫ a and b are called the limits of integration f(x), the thing that is being integrated is called the integrand dx indicates that the independent variable is x. The result of definite integration ∫ a b f(x)dx is a number. It is a fixed value, not an expression or an equation. It is a fixed area, like 4 or 7/13. The sum that we use in the calculation of the integral is often called a Reimann Sum ∑ i=1 n f(x i ∗ )Δx the sum without a limit Theorem: If f is continuous on [a,b], or if f has only a finite number of jump discontinuities, then f is integrable on [a,b]. ∫ 1 2 f(x)dx 180 Guided Notes | 5.2: The Definite Integral When the function I'm looking at is non-negative, f(x)≥0: The Riemann sum is the sum of the areas of a finite number of rectangles: rectangles approximating the area under a curve And the definite integral is the exact area under a curve from y=f(x) from a to b. area under a curve If the function I'm working with is both positive and negative... The Riemann sum is the sum of the areas of rectangles above the x-axis minus the sum of rectangles below the x-axis. rectangles above and below the xaxis approximating the area under a curve that crosses the xaxis f(x i ∗ )Δx ↑ negative positive The negative of the area of the rectangle. f(x i ∗ )Δx ↑ positive positive The area of the rectangle. And the definite integral is the net area (The sum of areas above the x-axis minus the sum of areas below the x-axis). net area under a curve that crosses the xaxis, with areas above shaded differently from areas below ∫ a b f(x)dx: Net area ∫ a b ∣f(x)∣dx: total area 1180 Guided Notes | 5.2: The Definite Integral Example: Express the limit shown below as a definite integral on the interval [1,5] lim n→∞ ∑ i=1 n 1+x i e x i Δx ∫ 1 5 1+x e x dx ∫ a b f(x)dx a=1b=5 Example: For f(x)=x 3 −6x, evaluate the Riemann sum by taking sample points to be the right endpoints and a=0, b=3 and n=6. Finite. Not taking limit yet. Δx= n b−a = 6 3−0 = 6 3 = 2 1 List of all interval endpoints: x 0 =0 x 1 = 2 1 x 2 =1 x 3 = 2 3 x 4 =2 x 5 = 2 5 x 6 =3←b i=1 i=2 i=3 i=4 i=5 i=6 Right endpoints R 6 =∑ i=1 6 f(x i )Δx Reiman sum with n=6 Δx= 2 1 = 2 1 [f( 2 1 )+f(1)+f( 2 3 )+f(2)+f( 2 5 )+f(3)] = 2 1 [−2.875+(−5)+(−5.625)+(−4)+(0.625)+9] =−3.8125 =−7.85← This is negative because the area below the x-axis is greater than the area above the x-axis.