Suppose g(x) is given by the following...

Transcript

00:01 Consider this graph of g of x below.

00:02 For this problem, we want to find the following definite integral.

00:06 So the first one is the integral from 0 to 3 of g of x, dx.

00:13 Now, this is just the area of the region bounded by g of x and the x axis on the closed interval 0 to 3.

00:22 So if i'm to shade this region, it should be this part of the graph.

00:27 Now looking at the shaded region to determine the area, we can separate this into three parts.

00:35 So the first region will be this region.

00:38 It's called it a sub 1, and this is our a sub 2, and then this is our a sub 3.

00:44 So this is equal to a1 plus a2 plus a3.

00:51 Now for a1, that's the area of a trapezoid whose upper base is equal to this distance, let's call it b1, lower base equal to this distance, it's called it b2, and height equal to this distance, let's call it h.

01:10 So it'll be one -half times a sum of b -1 and b -2, that's a negative 1 plus a negative 3 times a height of 1.

01:24 And then plus you have a2.

01:29 A2 is the area of the rectangle whose width is equal to 1.

01:35 So 1 times the height, which is, or the length, which is equal to negative 3, plus a3, which is a trapezoid as well.

01:47 So that's 1 half times the sum of upper base, which is this distance, let's call it b1.

01:56 And lower base equal to this distance, let's call it b2...

Question

Suppose g(x) is given by the following graph. y 1.0 0.5 1 2 3 4 5 6 -0.5 -1.0 Compute each of the following values: $\int_0^1 g(x) dx =$ $\int_1^6 g(x) dx =$ $\int_0^6 g(x) dx = $

Question

Suppose g(x) is given by the following graph. y 1.0 0.5 1 2 3 4 5 6 -0.5 -1.0 Compute each of the following values: $\int_0^1 g(x) dx =$ $\int_1^6 g(x) dx =$ $\int_0^6 g(x) dx = $

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