Math Problem Statement
Solution
Let's break this problem into steps to solve each part systematically.
(a) Sketch the function and find the area under the graph over [0, 4]
The function is a straight line that passes through the origin. Over the interval , the graph is a line starting at and ending at .
The area under the graph from is the area of the triangle formed by the x-axis, the line , and the function. The formula for the area of a triangle is: Here, the base is and the height is also . Thus:
Result for part (a): The area under the graph is 8.
(b) Right-endpoint approximation
For the right-endpoint approximation, divide the interval into rectangles. The width of each rectangle is: The right endpoints are: The area of each rectangle is . The total area is: Simplify: Using the formula for the sum of the first integers: we get: Take the limit as :
Result for part (b): The area under the graph using right-endpoint approximation is 8.
(c) Left-endpoint approximation
For the left-endpoint approximation, divide the interval into rectangles. The width of each rectangle is: The left endpoints are: The area of each rectangle is . The total area is: Simplify: Using the formula for the sum of the first integers: we get: Take the limit as :
Result for part (c): The area under the graph using left-endpoint approximation is 8.
Final Results:
- (a) Exact area: .
- (b) Right-endpoint approximation: .
- (c) Left-endpoint approximation: .
Do you want me to clarify any part of this?
Here are five related questions to consider:
- How does the midpoint approximation compare to the left- and right-endpoint approximations?
- Can you apply the trapezoidal rule to this problem? What result would you get?
- Why does the exact area match the limit of and ?
- How does increasing affect the accuracy of these approximations?
- Can you generalize this method to find the area under over any interval ?
Tip: When solving for approximations using sums, always simplify the summation expressions step by step to avoid errors!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Integral Calculus
Riemann Sums
Limits
Formulas
Area of a triangle: A = (1/2) * base * height
Sum of the first N integers: ∑(j=1 to N) j = N(N+1)/2
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Undergraduate Calculus
Related Recommendation
Right Endpoint Rule Area Approximation for f(x) = x^2 on [0,2]
Estimate Area Under Function f(x) = 4x + 1 Using Right and Left Endpoint Approximations
Estimating Area Under f(x) = 5 + x^2 Using Right Endpoints
Riemann Sum Approximation for Polynomial Function 4x^2 - x - 1
Right Endpoint Approximation R4 for f(x) = 4 - x over [0,4] and Limit as n Approaches Infinity