Question

4. (5 points) Sketch a graph and use geometry to evaluate the definite integral ∫

5
10
​
(2x−3)dx. Do not use the Fundamental Theorem of Calculus for this problem.

Answer 1

Explanation:

Above is the sketch of the graph. Remember the intergeal of a functions gives us area below the curve.

So, we want to find area between the x axis(x=0) and the function

The red segment is the function (2x-3)

The blue and green segment are the bounds of integration (5,10)

Notice that the region forms a trapezoid.

The area of a trapezoid is

`(b _(1) + b _(2) )/(2) * h`

B1 is The Blue Segment which is 7 units long

B2 is the green segment which is 17 units long

The height is the side perpendicular to the bases, which is 5.

So

`(7 + 17)/(2) * 5 = 60`

So the

∫ (2x-3) dx from. 5 to 10 =60