NUST NET (Engineering) · Maths · Integration. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.
A definite integral has limits and gives a number: integrate, substitute the upper limit, then subtract the value at the lower limit. No constant c is needed. Its properties often give the answer with almost no working, so look at the limits and the symmetry of the function before integrating.
The integral gives area with a sign: parts of the curve below the x-axis count as negative. For a real area, split the interval where the curve crosses the axis and add the sizes. Between two curves, integrate upper minus lower between their intersection points.
A separable differential equation is solved by moving all y terms to one side and all x terms to the other, then integrating both sides.
Integrating straight across a point where the curve crosses the x-axis when the question asks for area. The positive and negative parts cancel, so split the integral at the crossing and add the sizes of the parts.
Answer: 9/2 square units
Try each one, then open the answer.
Answer: D. y dy = x dx gives y2/2 = x2/2 + k, i.e. y2 − x2 = C. At (0, 2), C = 4. The circle would come from dy/dx = −x/y.
Answer: D. x5 is odd, so its integral from −1 to 1 is 0. For the even part, ∫−11 3x2 dx = [x3] from −1 to 1 = 1 − (−1) = 2.
Answer: A. The curve cuts the x-axis at x = ±2: ∫−22 (4 − x2) dx = 16 − 16/3 = 32/3. 16/3 is only the half from 0 to 2.
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