Integral Calculus

Question 1

Find an equation of the curve passing through the point (3, 2) and having slope 5x² - x + 1 at every point (x, y).

Answer 1

y = 5/3x³ + 1/2x² + x - 83/2

Question 2

Find the area under the parabola y = x² - 2x + 2, above the x-axis, and between x = 0 and x = 1.

a) 1/2 units²
b) 4/3 units²
c) 5/6 units²
d) 3/2 units²

Answer 2

b) 4/3 units²

Question 3

Find the area of the region between the parabola y = x² and the line y = 1/2x + 2.

a) 3.95 units²
b) 4.62 units²
c) 5.95 units²
d) 6.21 units²

Answer 3

a) 3.95 units²

Question 4

Find the area of the region between the curves y = sin x and y = cos x from x = 0 to x = π / 4.

a) 0.31 units²
b) 0.41 units²
c) 0.51 units²
d) 0.61 units²

Answer 4

b) 0.41 units²

Question 5

Find the centroid of the region bounded by the parabola y = x² and the line y = 1/2x + 2.

a) (0.25, 1.30)
b) (0.75, 1.50)
c) (0.25, 3.10)
d) (0.75, 5.10)

Answer 5

a) (0.25, 1.30)

Question 6

Find the moment of inertia of the region bounded by the parabola y = x² and the line y = 1/2x + 2.

a) 8.42 units²
b) 8.92 units²
c) 8.36 units²
d) 8.23 units²

Answer 6

d) 8.23 units²

Question 7

Find the arc length of the curve y = x³/₂ from x = 0 to x = 5.

a) 11.56 units
b) 12.41 units
c) 13.72 units
d) 14.22 units

Answer 7

b) 12.41 units

Question 8

Find the volume of the solid obtained by revolving, about the y-axis, the region in the first quadrant inside the circle x² + y² = 5².

a) 250 π
b) 250/2 π
c) 250/3 π
d) 250/4 π

Answer 8

c) 250/3 π

Question 9

Find the volume of the solid obtained by revolving, about the y-axis, the region in the first quadrant bounded by the parabola y = 2 - x² and above the parabola y = x².

a) π
b) π/2
c) π/4
d) π/8

Answer 9

a) π

Question 10

Find the area of the surface of revolution generated by revolving, about the x-axis, the arc of parabola y² = 12x from x = 0 to x = 3.

a)117.86 units²
b) 127.86 units²
c) 137.86 units²
d) 147.86 units²

Answer 10

c) 137.86 units²