A graph of f(x) is shown above. Using the geometry of the...

A graph of f(x) is shown above. Using the geometry of the graph, evaluate the definite integrals. a) ??³ f(x) dx = b) ??? f(x) dx = c) ??? f(x) dx = d) ??¹? f(x) dx = e) ??¹? f(x) dx =

Transcript

00:01 So in this question, they give us a graph of a function f of x shown above, and using the geometry of the graph, we want to evaluate each of the following definite intervals.

00:12 In a, i want the integral from 0 to 2 of f of x dx.

00:19 Remember, a definite interval computes the net -signed area between my function and the x -axis over the appropriate interval.

00:30 So on the interval from 0 to 2, i'm looking at the area of this rectangle.

00:41 Now, this rectangle has a width of 2, and it's got a height of 3, 4, 5, 6, 7, 8, 9.

00:50 So i have an area of 18 here.

00:54 However, that area lies below the x -axis.

00:58 Therefore, that definite integral is counted as negative by the definite interval, giving me negative 18.

01:09 In b, i want the interval from 2 to 4 of f of x d.

01:14 So now i'm looking at again my net side area between my function and the horizontal axis over that interval from x equals 2 to x to x equals 4.

01:29 The area of this triangle, one half base times height, is one half, my base is two, my height is nine, i'm getting nine.

01:42 However, again, that area is below the x -axis, and so that area is counted as negative by the definite interval.

01:51 My integral from 2 to 4 of f of x, dx, is negative 9.

01:57 Then in c, i want the integral from 0 to 4 of f of x d x.

02:04 To do that, i just add the integral from 0 to 2 to the integral from 2 to 4, giving me negative 27...

Question

A graph of f(x) is shown above. Using the geometry of the graph, evaluate the definite integrals. a) ??³ f(x) dx = b) ??? f(x) dx = c) ??? f(x) dx = d) ??¹? f(x) dx = e) ??¹? f(x) dx =

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