Find the total area of the region bounded by the curve y = 3x^2 - 6x, the x-axis, and the … — JAMB UTME

Find the total area of the region bounded by the curve $y = 3x^2 - 6x$ , the $x$ -axis, and the vertical lines $x = 0$ and $x = 3$ .

Answer: 8

The total area bounded by the curve and the x-axis between x = 0 and x = 3 is 8 square units.

To find the total geometric area bounded by a curve and the x-axis, we must split the integral at any x-intercepts within the domain. For

y = 3x^2 - 6x

, the x-intercepts are

x = 0

and

x = 2

. Between

x = 0

and

x = 2

, the curve lies below the x-axis, giving an area magnitude of

4

. Between

x = 2

and

x = 3

, the curve lies above the x-axis, giving an area magnitude of

4

. Summing these positive magnitudes gives a total area of

8

Find the roots of the curve

y = 3x^2 - 6x

within the given interval

[0, 3]

Setting

3x^2 - 6x = 0

yields

3x(x - 2) = 0

, giving

x = 0

and

x = 2

Roots inside the integration boundaries indicate where the curve crosses the x-axis, changing the sign of

y

Determine the position of the curve relative to the x-axis on each sub-interval.

[0, 2]

y \le 0

(below the x-axis). On

[2, 3]

y \ge 0

(above the x-axis).

Geometric area must be non-negative, so regions below the x-axis require integrating

-y

or taking the absolute value of the integral.

Evaluate the area

A_1

for the region below the x-axis from

x = 0

x = 2

A_1 = \int_{0}^{2} (6x - 3x^2) \, dx = \left[ 3x^2 - x^3 \right]_{0}^{2} = (3(4) - 8) - 0 = 4

Integrating

-y = 6x - 3x^2

yields the positive magnitude of the area below the x-axis.

Evaluate the area

A_2

for the region above the x-axis from

x = 2

x = 3

A_2 = \int_{2}^{3} (3x^2 - 6x) \, dx = \left[ x^3 - 3x^2 \right]_{2}^{3} = (3^3 - 3(3^2)) - (2^3 - 3(2^2)) = 0 - (-4) = 4

Direct integration of

y

[2, 3]

gives the area above the x-axis.

Combine the areas of both sub-regions.

Total Area

= A_1 + A_2 = 4 + 4 = 8

The total geometric area is the sum of the magnitudes of the areas of all separate bounded regions.

Calculating area under curves crossing the x-axis by splitting definite integrals at real roots

Question