Q5. Find the area under y = √x from x = 0 to x = 9.
∫09 x1/2 dx = (2/3) × 93/2 = (2/3) × 27 = 18.
Q6. Find the area between the curve y = x2 − 1, the x-axis and the ordinates x = 0 and x = 2.
The curve is below the axis on [0, 1] and above it on [1, 2]. ∫01 (x2 − 1) dx = −2/3, an area of 2/3; ∫12 (x2 − 1) dx = 4/3. The area is 2/3 + 4/3 = 2. (The plain integral from 0 to 2 is 2/3.)
Q7. Find the area of the region between the curve x = y2 + 1, the y-axis and the lines y = 0 and y = 3.
Use horizontal strips: ∫03 (y2 + 1) dy = 9 + 3 = 12.
Q8. Find the area of the ellipse 3x2 + 4y2 = 12.
Dividing by 12 gives x2/4 + y2/3 = 1, so a = 2 and b = √3, and the area is π × 2 × √3 = 2√3π.
Q9. Find the area of the quarter of the circle x2 + y2 = 9 in the first quadrant by horizontal strips.
x = √(9 − y2), so the area is ∫03 √(9 − y2) dy = (9/2)(π/2) = 9π/4, a quarter of 9π.
Q10. Find the area under y = 1/x from x = 1 to x = e, and under y = ex from x = 0 to x = 1.
log e − log 1 = 1, and e1 − e0 = e − 1.