
William W. answered 06/02/23
Experienced Tutor and Retired Engineer
The slope of the tangent line is the derivative. Using the power rule, the derivative is y' = 10x. At x = 3, that slope is 30. Using the point slope form, the tangent line at (3. 45) is y - 45 = 30(x - 3) or y = 30x - 45.
Graphing these we get:
So the area under the curve from 0 to 1.5 is 0∫1.5 5x2 dx
And the area under the curve from 1.5 to 3 is: 1.5∫3 (5x2 - (30x - 45) dx
You can do this manually by finding the antiderivative of each or you can use your calculator and you get an area of 11.25 square units