Determine the following integrals: 9.1.1 ∫ −4 dt 9.1.2 ∫ x^5 (x^3 − 9x^2) dx 9.2 The diagram below shows function f defined by f(x) = -x^2 + 2x + 3 The graph of f cuts the x-axis at A(-1; 0) and B(3; 0) and the y-axis at point C(0; 3) - NSC Technical Mathematics - Question 9 - 2023 - Paper 1

First, simplify the integrand:

$$∫ x^5 (x^3 - 9x^2) dx = ∫ (x^8 - 9x^7) dx.$$

Now calculate the integral term by term:

= rac{x^9}{9} - rac{9x^8}{8} + C

Hence, the result of the integral is

rac{x^9}{9} - rac{9x^8}{8} + C.

To find the total shaded area bounded by the curve and the x-axis, we calculate:

1. Area AOB: $$$$

A_{AOB} = ∫_{-1}^{3} (-x^2 + 2x + 3) dx

$$Step by step: - Evaluate the indefinite integral:$$

rac{x^3}{3} + x^2 + 3x

$$- Calculate from -1 to 3:$$

A_{AOB} = [rac{3^3}{3} + 3^2 + 9] - [rac{(-1)^3}{3} + (-1)^2 + 3(-1)]

$$Simplifying results in:$$

= rac{9 + 9 + 9}{1} + rac{1}{3} + 1 - 3 = 10.67 ext{ units}^2.

$$2. **Area of the triangle CEF** (with vertices at points C, E, E): For E(2; 0) and C(0; 3): Base = 2 and Height = 3.$$

A_{CEF} = rac{1}{2} imes base imes height = rac{1}{2} imes 2 imes 3 = 3 ext{ units}^2.

$$3. **Total shaded area**:$$

A_{Total} = A_{AOB} + A_{CEF} = 10.67 + 3 = 13.67 ext{ units}^2.

$$$$