Math Problem Statement
Solution
The problem asks for two things:
- Set up the definite integral for the area between the curve and the x-axis over the interval .
- Approximate the area using the left sum with the given partition .
Step 1: Set up the definite integral
The function given is . The definite integral for the area between the graph of and the x-axis over the interval is:
Step 2: Approximate the area using the left sum
The left sum approximation uses the left endpoints of each subinterval defined by the partition . These subintervals are:
- From to
- From to
- From to
For each subinterval, the left endpoint is used to evaluate (i.e., ) and these values are multiplied by the width of the corresponding subinterval.
Step-by-step calculation of the left sum:
-
Width of each subinterval:
-
Evaluating the function at the left endpoints:
-
Left sum approximation: Substituting the values: Factoring out the 4:
Let me calculate this approximation.The left sum approximation for the area is:
Would you like more details about the process, or do you have any further questions?
Here are five related questions to explore this topic further:
- How would the right sum approximation differ from the left sum?
- Can you use the midpoint Riemann sum to approximate the area?
- How does increasing the number of partitions affect the accuracy of the approximation?
- What is the exact value of the definite integral ?
- How does the trapezoidal rule compare to the left sum for approximating integrals?
Tip: When using a left sum approximation, the more partitions you have, the closer your approximation will be to the actual area under the curve.
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Math Problem Analysis
Mathematical Concepts
Definite Integrals
Riemann Sums
Square Root Functions
Formulas
Definite Integral: A = ∫_a^b f(x) dx
Left Riemann Sum: A_left = Σ f(x_i) * Δx_i
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12 or Early College
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