Question

Find the area bounded by the curve y = x 1/2 + 2, the x-axis, and the lines x = 1 and x = 4 A. 16 B. 10 2/3 C. 7 1/2 D. 28 1/2

Answer 1

To find the area of the curve subject to these constraints, we must take the integral of y = x ^ (1/2) + 2 from x=1 to x=4
Take the antiderivative: Remember that this what the original function would be if our derivative was x^(1/2) + 2 
antiderivative (x ^(1/2) + 2) =  (2/3) x^(3/2) + 2x
* To check that this is correct, take the derivative of our anti-derivative and make sure it equals x^(1/2) + 2

To find integral from 1 to 4:
Find anti-derivative at x=4, and subtract from the anti-derivative at x=1
2/3 * 4 ^ (3/2) + 2(4) - (2/3) *1 - 2*1
2/3 (8) + 8 - 2/3 - 2                               Collect like terms
2/3 (7) + 6                                             Express 6 in terms of 2/3
2/3 (7) + 2/3 (9)
2/3 (16) = 32/3 = 10 2/3          Answer is B