Question

for each problem, write as a definite integral and sketch the integral represented. do not evaluate the integral

1. ( y = sqrt { x } ; 3,6 ) and the x - axis
2. ( y = x ^ { 2 } - 4 ; - 1,1 ) and the x - axis

Explanation:

Problem 4

Step1: Identify bounds and function

Interval is [3,6], function is $y=\sqrt{x}$ (non-negative here).

Step2: Write definite integral

Integral of function over interval: $\int_{3}^{6} \sqrt{x} dx$

Step1: Check function sign on [-1,1]

$y=x^2-4$ is negative (since $x^2\leq1$, so $x^2-4\leq-3$).

Step2: Write integral for area (absolute value)

Integral of absolute value of function: $\int_{-1}^{1} |x^2 - 4| dx$

Answer:

$\int_{3}^{6} \sqrt{x} dx$

Problem 5