Approximate the area under the graph of f(x) over the specified interval by dividing the interval into the indicated number of subintervals and using the left endpoint of each subinterval

Question

f(x) =-0.02x4- 0.2x2+ 10; interval [-4, 4]; 4 subintervals

Answer 1

there're 4 intervals, from -4<x<-2, -2<x<0, 0<x<2, to 2<x<4

each interval has base of length 2

calculate f(x) for the height, using the left boundaries: f(-4), f(-2), f(0) and f(2)

the 4 areas are 2f(-4), 2f(-2), 2f(0), and 2f(2)

sum them to get the estimate of the area under the curve

factor out 2: 2( f(-4) + f(-2) + f(0) + f(2) )

check the answer by taking the integral evaluated for limits -4 to 4

integral of f(x) = .(.02/5)x^5 - (.2/3)x^3 + 10x + c (but ignore the c as it cancels out when evaluated with limits of integration)

f(x) = .02x^4 - .2x^2 + 10

let z=x^2, then f(z) = .02z^2 -2z + 10 which is a very flat nearly horizontal parabola with a y intercept of 10, so f(x) = 10 or very close to 10

10 times 4 = 40 = a close estimate of the area under the curve. It's about 39 to 41

calculate f(x) for x= -4, -2, 0 and 2, then sum them, for the interval heights, then multiply by 2 the base for each interval

that gives a close approximation for the area between the curve and the x axis and between x=-4 and x=4