Podcast
Questions and Answers
When approximating the area under a curve using rectangles, which method generally provides the most accurate estimate?
When approximating the area under a curve using rectangles, which method generally provides the most accurate estimate?
- Left endpoint rule
- They all provide the same accuracy.
- Right endpoint rule
- Midpoint rule (correct)
As the number of rectangles used in approximating the area under a curve increases, what happens to the accuracy of the approximation?
As the number of rectangles used in approximating the area under a curve increases, what happens to the accuracy of the approximation?
- It increases (correct)
- It decreases
- It stays the same
- It fluctuates randomly
If a function is increasing over an interval, which method will overestimate the area under the curve?
If a function is increasing over an interval, which method will overestimate the area under the curve?
- Left endpoint rule
- Midpoint rule
- Right endpoint rule (correct)
- All of the above
Consider a curve $y = f(x)$ on the interval $[a, b]$. If $\Delta x = \frac{b-a}{n}$, what does $n$ represent in the context of Riemann sums?
Consider a curve $y = f(x)$ on the interval $[a, b]$. If $\Delta x = \frac{b-a}{n}$, what does $n$ represent in the context of Riemann sums?
When using the left endpoint rule to approximate the area under a curve, what value determines the height of each rectangle?
When using the left endpoint rule to approximate the area under a curve, what value determines the height of each rectangle?
What is the primary difference between using the midpoint rule and the trapezoidal rule for approximating the area under a curve?
What is the primary difference between using the midpoint rule and the trapezoidal rule for approximating the area under a curve?
Suppose you are approximating the area under a curve using rectangles of equal width. If you double the number of rectangles, what is the effect on the width of each individual rectangle?
Suppose you are approximating the area under a curve using rectangles of equal width. If you double the number of rectangles, what is the effect on the width of each individual rectangle?
Under what condition would both the left and right endpoint rules give the exact same approximation for the area under a curve?
Under what condition would both the left and right endpoint rules give the exact same approximation for the area under a curve?
How does increasing the number of rectangles affect the difference between the approximations obtained by the left and right endpoint rules?
How does increasing the number of rectangles affect the difference between the approximations obtained by the left and right endpoint rules?
What is a definite integral's relationship to approximating the area using rectangles?
What is a definite integral's relationship to approximating the area using rectangles?
Flashcards
Area Approximation
Area Approximation
Estimating the area under a curve within an interval by dividing the area into rectangles and summing their areas.
Left Endpoint Rule
Left Endpoint Rule
Using the left endpoint of each subinterval to determine the height of the rectangle.
Midpoint Rule
Midpoint Rule
Using the midpoint of each subinterval to determine the height of the rectangle.
Right Endpoint Rule
Right Endpoint Rule
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Study Notes
- It is possible to estimate the area under a curve on a given interval using rectangles.
- To approximate the area under a curve on a given interval using n rectangles, use left endpoint, midpoint, and right endpoint evaluation rules.
Approximating Area with 16 Rectangles and endpoints for y = x² + 1 on [0, 1]
- There are 16 rectangles, with evaluation points given by cᵢ = iΔx, where i ranges from 0 to 15.
- A₁₆ = Δx Σ f(cᵢ) where i ranges from 0 to 15
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- The left endpoint approximation is ≈ 1.3027, which gives an underestimation of the area.
- The midpoint approximation is ≈ 1.3330.
- There are 16 rectangles, with evaluation points given by cᵢ = iΔx + Δx/2 where i ranges from 0 to 15.
- A₁₆ = Δx Σ f(cᵢ) where i ranges from 0 to 15
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- The right endpoint approximation is ≈ 1.3652, which gives an overestimation of the area.
- There are 16 rectangles, with evaluation points given by cᵢ = iΔx + Δx where i ranges from 0 to 15.
- A = 4/3 ≈ 1.33333333 is the exact area.
- A₁₆ = Δx Σ f(cᵢ) where i ranges from 0 to 15
Approximating Area with 16 Rectangles and endpoints for y = x² + 1 on [0, 2]
- There are 16 rectangles, with evaluation points given by cᵢ = iΔx, where i ranges from 0 to 15.
- A₁₆ = Δx Σ f(cᵢ) where i ranges from 0 to 15 ≈ 4.4219
- There are 16 rectangles, with evaluation points given by cᵢ = iΔx + Δx/2 where i ranges from 0 to 15.
- There are 16 rectangles, with evaluation points given by cᵢ = iΔx + Δx where i ranges from 0 to 15.
- A₁₆ = Δx Σ f(cᵢ) where i ranges from 0 to 15 ≈ 4.6640
- A₁₆ = Δx Σ f(cᵢ) where i ranges from 0 to 15 ≈ 4.9219
Approximating Area with 16 Rectangles and endpoints for y = √x + 2 on [1, 4]
- There are 16 rectangles, with evaluation points given by cᵢ = 1 + iΔx, where i ranges from 0 to 15.
- A₁₆ = Δx Σ f(cᵢ) where i ranges from 0 to 15 ≈ 6.2663
- There are 16 rectangles, with evaluation points that are midpoints given by cᵢ = 1 + iΔx + Δx/2 where i ranges from 0 to 15.
- A₁₆ = Δx Σ f(cᵢ) where i ranges from 0 to 15 ≈ 6.3340
- There are 16 rectangles, with evaluation points that are the right endpoints given by cᵢ = 1 + iΔx, where i ranges from 1 to 16.
- A₁₆ = Δx Σ f(cᵢ) where i ranges from 1 to 16 ≈ 6.4009
Approximating Area with 16 Rectangles and endpoints for y = e⁻²ˣ on [-1, 1]
- There are 16 rectangles, with evaluation points as the left endpoints given by cᵢ = -1 + iΔx - Δx where i ranges from 1 to 16.
- A₁₆ = Δx Σ f(cᵢ) where i ranges from 1 to 16 ≈ 4.0991
- There are 16 rectangles, with evaluation points as the midpoints given by cᵢ = -1 + iΔx - Δx/2 where i ranges from 1 to 16.
- A₁₆ = Δx Σ f(cᵢ) where i ranges from 1 to 16 ≈ 3.6174
- There are 16 rectangles, with evaluation points as the right endpoints given by cᵢ = -1 + iΔx where i ranges from 1 to 16.
- A₁₆ = Δx Σ f(cᵢ) where i ranges from 1 to 16 ≈ 3.1924
Approximating Area with 50 Rectangles and endpoints for y = cos x on [0, π/2]
- There are 50 rectangles, with evaluation points given by cᵢ = iΔx where i is from 0 to 49.
- A₅₀ = Δx Σ f(cᵢ) where i ranges from 0 to 49 ≈ 1.0156
- There are 50 rectangles, with evaluation points given by cᵢ = Δx/2 + iΔx where i is from 0 to 49.
- A₅₀ = Δx Σ f(cᵢ) where i ranges from 0 to 50 ≈ 1.00004
- There are 50 rectangles, with evaluation points given by cᵢ = Δx + iΔx where i is from 0 to 49.
- A₅₀ = Δx Σ f(cᵢ) where i ranges from 0 to 49 ≈ 0.9842
Approximating Area with 100 Rectangles and endpoints for y = x³ - 1 on [-1, 1]
- There are 100 rectangles and the evaluation points are left endpoints which are given by cᵢ = -1 + iΔx – Δx where i is from 1 to 100.
- A₁₀₀ = Δx Σ f(cᵢ) where i ranges from 1 to 100 ≈ -2.02
- There are 100 rectangles and the evaluation points are midpoints which are given by cᵢ = -1 + iΔx Δx/2 where i is from 1 to 100.
- A₁₀₀ = Δx Σ f(cᵢ) ≈ -2 where i ranges from 1 to 100
- There are 100 rectangles and the evaluation points are right endpoints which are given by cᵢ = -1 + iΔx where i is from 1 to 100.
- A₁₀₀ = Δx Σ f(cᵢ) where i ranges from 1 to 100 ≈ -1.98
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