Leave your answer in algebra form.

Question

(1) What is the area below

y=−x2+4 and above the x-axis?

(2)The position of a particle is represented by f(t) = 4t3 – 3t + 15

What is its instantaneous velocity at time t = 3?

Answer 1

Draw the graph. The region is bounded above by the graph of y = 4 - x² and below by the x-axis between x = -2 and x = 2. The region is symmetric with respect to the y-axis, so we can just find the area of the region between x = 0 and x = 2 and then double the result.

Area of region = 2∫_{(0 to 2)} (4 - x²)dx = 2(4x - (1/3)x³)_{(0 to 2)} = 2 [ (8 - 8/3) - (0) ] = 32/3

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f(t) = 4t³ - 3t + 15

f'(t) = 12t² - 3

Instantaneous velocity when t = 3 is f'(3) = 105