Definite Integral Area 30Db1F

1. **State the problem:** We need to find the area under the curve defined by the function $$f(x) = -0.2x^2 + 2.8x - 1.8$$ from $$x=1$$ to $$x=13$$ using the definite integral $$\int_1^{13} (-0.2x^2 + 2.8x - 1.8) \, dx$$. 2. **Recall the integral formula:** The definite integral of a polynomial function $$ax^n$$ is $$\int ax^n \, dx = \frac{a}{n+1}x^{n+1} + C$$. 3. **Integrate the function:** $$\int (-0.2x^2 + 2.8x - 1.8) \, dx = \int -0.2x^2 \, dx + \int 2.8x \, dx - \int 1.8 \, dx$$ Calculate each term: - $$\int -0.2x^2 \, dx = -0.2 \cdot \frac{x^3}{3} = -\frac{0.2}{3}x^3 = -0.0667x^3$$ - $$\int 2.8x \, dx = 2.8 \cdot \frac{x^2}{2} = 1.4x^2$$ - $$\int 1.8 \, dx = 1.8x$$ So the antiderivative is: $$F(x) = -0.0667x^3 + 1.4x^2 - 1.8x + C$$ 4. **Evaluate the definite integral:** $$\int_1^{13} (-0.2x^2 + 2.8x - 1.8) \, dx = F(13) - F(1)$$ Calculate $$F(13)$$: $$F(13) = -0.0667 \times 13^3 + 1.4 \times 13^2 - 1.8 \times 13$$ $$= -0.0667 \times 2197 + 1.4 \times 169 - 23.4$$ $$= -146.47 + 236.6 - 23.4 = 66.73$$ Calculate $$F(1)$$: $$F(1) = -0.0667 \times 1 + 1.4 \times 1 - 1.8 \times 1 = -0.0667 + 1.4 - 1.8 = -0.4667$$ 5. **Subtract to find the area:** $$F(13) - F(1) = 66.73 - (-0.4667) = 66.73 + 0.4667 = 67.20$$ 6. **Round the answer:** The problem asks to round to the nearest whole number. $$\boxed{67}$$ **Note:** The user provided answer was 81.84, but the correct calculation yields approximately 67.