Math 233 Unit 5 Integration Individual Project Assignment

Question

Page 1 Of 2math233 Unit 5 Integrationindividual Project Assignment V Evaluate the following integrals: $\int (5x^4 - 2x^2 + 2x) \, dx$ $\int \left(\frac{1}{\sqrt{x^3}} + 5x^5 \right) dx$ $\int \frac{1}{7x} \, dx$ $\int (9x^2 - 8x + 1) \, dx$ $\int \frac{1}{(x+5)^3} dx$ Consider the graph below, and answer the following: Set up the definite integral to calculate the area of the shaded region on the graph, where the red curve is the graph of the function $f(x) = 0.5x^2 + x + 2$, for $1 Find the area of the shaded region. The velocity (in feet per second) of a moving object at various times \(t$ (in seconds) has been recorded and is modeled by the function $v(t) = 9t^2 + 4t$, with an initial position of 0 feet. Find the position function $s(t)$, where $s$ is in feet. Find the position of the object after 2 seconds. Find the change in position of the object between 5 and 10 seconds. Suppose you are tasked with modeling a city’s population growth. The growth rate is given by $r(t) = 200 \times 1.075^t$, where $t$ is years from 2010. The current population is 75,000. Choose three values of $t$ between 5 and 20. Find the function $P(t)$ that models the population growth. Complete a table with the three chosen $t$ values, calculating the corresponding population $P(t)$. Interpret the meaning of the integral $\int 200 \times 1.075^t \, dt$ in the context of population modeling.

Dr. Jack HW Helper · Accepted Answer

This comprehensive mathematical analysis explores a sequence of integral calculus problems, applied to various real-world contexts such as area calculation, motion, and population growth modeling. Each problem demonstrates different integral concepts, from indefinite integrals to applications involving physics and demographic analysis. Through detailed explanations and calculations, this paper aims to provide clarity on the use of integration in varied scenarios, emphasizing the importance of understanding integral setup, calculation, and interpretation. Evaluation of Integrals The indefinite integrals incorporate polynomial, rational, and algebraic functions. The first integral, $\int (5x^4 - 2x^2 + 2x) \, dx$, involves straightforward power rule integration. Integrating term-by-term yields: $$ \frac{5}{5} x^{5} - \frac{2}{3} x^{3} + x^{2} + C = x^{5} - \frac{2}{3} x^{3} + x^{2} + C. $$ The second integral, $\int \left(\frac{1}{\sqrt{x^3}} + 5x^5 \right) dx$, involves simplifying the first term: $$ \frac{1}{\sqrt{x^3}} = x^{-\frac{3}{2}}. $$ Applying power rule: $$ \int x^{-\frac{3}{2}} dx = \frac{x^{-\frac{1}{2}}}{-\frac{1}{2}} = -2 x^{-\frac{1}{2}} + C. $$ For the second term: $$ \int 5x^5 dx = 5 \times \frac{x^{6}}{6} = \frac{5}{6} x^{6} + C. $$ Hence, the entire integral evaluates to: $$ -2 x^{-\frac{1}{2}} + \frac{5}{6} x^6 + C. $$ The third integral: $$ \int \frac{1}{7x} dx = \frac{1}{7} \int \frac{1}{x} dx = \frac{1}{7} \ln |x| + C. $$ The fourth integral: $$ \int (9x^2 - 8x + 1) dx = 3 x^3 - 4 x^2 + x + C. $$ The fifth integral: $$ \int \frac{1}{(x+5)^3} dx = -\frac{1}{2 (x+5)^2} + C, $$ where I used substitution $u = x + 5$, $du = dx$, leading to $\int u^{-3} du = - \frac{1}{2} u^{-2}$. Calculating Area Under a Curve To find the area of the shaded region bounded by the curve $f(x) = 0.5x^2 + x + 2$ over $[1, 5]$, we set up the definite integral: $$ A = \int_{1}^{5} (0.5x^2 + x + 2) dx. $$ Evaluating: \[ \int 0.5x^2 dx = \frac{0.5}{3} x^3 = \f

Math 233 Unit 5 Integration Individual Project Assignment

Page 1 Of 2math233 Unit 5 Integrationindividual Project Assignment V

Paper For Above instruction

Evaluation of Integrals

Calculating Area Under a Curve

Velocity and Position Analysis

Modeling Population Growth

Conclusion

References