Math Problem Statement
Use the Riemann sums to evaluate the integral from -3 to 4 of f(x), where f(x) is piecewise defined as 2x for 0 ≤ x ≤ 10, and x^2 + 2x - 4 for x ≤ 0.
Solution
The problem asks to evaluate the integral using the Riemann sum method, where is defined as:
\begin{cases} 2x & \text{for } 0 \leq x \leq 10 \\ x^2 + 2x - 4 & \text{for } x \leq 0 \end{cases}$$ To solve this integral using Riemann sums, follow these steps: 1. **Divide the interval $$[-3, 4]$$ into smaller subintervals**: Choose a certain number of subintervals $$n$$ and find the width of each subinterval $$\Delta x$$. 2. **Find sample points**: Choose sample points in each subinterval, which could be the left endpoint, right endpoint, or midpoint, depending on the type of Riemann sum you want to use (left, right, or midpoint sum). 3. **Evaluate $$f(x)$$ at the sample points**: Compute the function $$f(x)$$ at the chosen sample points within each subinterval. 4. **Sum up the areas of the rectangles**: Calculate the sum of the areas of the rectangles formed by the width $$\Delta x$$ and the height $$f(x)$$ at each sample point. For this integral, since $$f(x)$$ is piecewise-defined, the integral should be split into two parts: 1. **For the interval $$[-3, 0]$$**: $$\int_{-3}^{0} (x^2 + 2x - 4) \, dx$$ 2. **For the interval $$[0, 4]$$**: $$\int_{0}^{4} 2x \, dx$$ I will now evaluate these integrals: 1. $$\int_{-3}^{0} (x^2 + 2x - 4) \, dx$$: $$\int_{-3}^{0} x^2 \, dx + \int_{-3}^{0} 2x \, dx - \int_{-3}^{0} 4 \, dx$$ Evaluating each term separately: - $$\int_{-3}^{0} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{-3}^{0} = 0 - \left(-9\right) = 9$$ - $$\int_{-3}^{0} 2x \, dx = \left[ x^2 \right]_{-3}^{0} = 0 - 9 = -9$$ - $$\int_{-3}^{0} 4 \, dx = \left[ 4x \right]_{-3}^{0} = 0 - (-12) = 12$$ Therefore: $$\int_{-3}^{0} (x^2 + 2x - 4) \, dx = 9 - 9 + 12 = 12$$ 2. $$\int_{0}^{4} 2x \, dx$$: $$\left[ x^2 \right]_{0}^{4} = 16 - 0 = 16$$ The total integral is the sum of the two parts: $$\int_{-3}^{4} f(x) \, dx = 12 + 16 = 28$$ So, the value of the integral is 28. Would you like more details or have any questions about this solution? ### Related Questions: 1. What is a Riemann sum and how is it used to approximate definite integrals? 2. How do we choose sample points for a Riemann sum (left, right, or midpoint)? 3. What is the significance of breaking integrals for piecewise-defined functions? 4. How does the Fundamental Theorem of Calculus relate to evaluating definite integrals? 5. How would the result change if different intervals were used? **Tip**: When dealing with piecewise functions, ensure to split the integral at the points where the function definition changes.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Integral Calculus
Riemann Sums
Piecewise Functions
Formulas
∫ f(x) dx
Riemann sum formula
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12
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