Find the Area of the Region

Question

**Please show steps

Find the area of the region bounded by the parabola y=5(x^2), the tangent line to this parabola at (3,45) and the x-axis.

Answer 1

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

Answer 2

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