Determine the area under the curve y = 3x^2 - 3 above the x-axis between x = 1 and x = 3. — JAMB UTME

Determine the area under the curve $y = 3x^2 - 3$ above the $x$ -axis between $x = 1$ and $x = 3$ .

Answer: 20 square units

The area under the curve between

x = 1

and

x = 3

is 20 square units.

The area under the curve

y = 3x^2 - 3

from

x = 1

x = 3

is obtained by calculating the definite integral

\int_{1}^{3} (3x^2 - 3) \, dx = [x^3 - 3x]_{1}^{3} = (3^3 - 3(3)) - (1^3 - 3(1)) = 18 - (-2) = 20

Set up the definite integral representing the area under the curve.

A = \int_{1}^{3} (3x^2 - 3) \, dx

The area bounded by a non-negative curve

y = f(x)

, the

x

-axis, and vertical lines

x = a

and

x = b

is given by

\int_{a}^{b} f(x) \, dx

Integrate the polynomial function term by term.

\int (3x^2 - 3) \, dx = x^3 - 3x + C

Applying the power rule of integration:

\int 3x^2 dx = x^3

and

\int 3 dx = 3x

Apply the Fundamental Theorem of Calculus by substituting the limits of integration.

[x^3 - 3x]_{1}^{3} = (3^3 - 3(3)) - (1^3 - 3(1)) = 18 - (-2) = 20

Evaluating

F(b) - F(a)

gives

(27 - 9) - (1 - 3) = 18 - (-2) = 20

Definite Integrals and Area Under Curves

Question