Suppose f(x) 12 (a) The rectangles in the graph on the left illustrate a left endpoint Riemann sum for f(x) on the interval 3 x 5. The value of this left endpoint Riemann sum is and it the area of the region enclosed by y = f(x), the x-axis, and the vertical is an underestimate of lines a 3 and r = 5. = (b) The rectangles in the graph on the right illustrate a right endpoint Riemann sum for f(x) on the interval 3 5. The value of this right endpoint Riemann sum is and it the area of the region enclosed by y=f(x), the x-axis, and the vertical is an overestimate of lines a 3 and a = 5. The 5 rectangles in the graph below illustrate a midpoint interval [0, 2]. The value of this Riemann sum is BRASAHAS 16 15 14 13 12 11 10 9 8 7 Riemann sum for f(x) = 2* on the Riemann sum for y= 2 on [0, 2] The rectangles in the graph below illustrate a left endpoint Riemann sum for f(x) = The value of this left endpoint Riemann sum is region enclosed by y = f(x), the x-axis, and the vertical lines x = 2 and x = 6. 4 and it is equal to +2r on the interval [2, 6]. Left endpoint Riemann sum for y=+2r on [2,6 the area of the The rectangles in the graph below illustrate a right endpoint Riemann sum for f(x)= + 2x on the interval [2,6]. -2 4 the area The value of this right endpoint Riemann sum is of the region enclosed by y = f(x), the x-axis, and the vertical lines x = 2 and x = 6. and it is an equal to Right endpoint Riemann sum for y=+2r on [2,6) Using left and right Riemann sums based on the diagrams above, we definitively conclude that 4. S S 4 + 2x dx 45+ +2edes [7 2x 4 S + 2rd Hint: For the last integral, you should consistently choose either to underestimate or overestimate the area. This may require that you use the left Riemann sum for some x-intervals and the right Riemann sum for other x-intervals.