Area of each rectangle

Definition

The area of each rectangle refers to the individual contributions of small rectangles used to approximate the total area between curves. This concept is crucial in calculating the area between two curves by dividing the region into multiple rectangles and summing their areas, which can be refined using limits as the number of rectangles increases. Each rectangle's height is determined by the value of the function at specific points, while its width represents a small change in the x-direction.

5 Must Know Facts For Your Next Test

1. The area of each rectangle is calculated as the product of its height and width, expressed mathematically as Area = Height × Width.
2. When approximating areas between curves, rectangles can be drawn either above or below the curve, affecting the accuracy of the approximation.
3. As the number of rectangles increases (i.e., as their width decreases), the approximation becomes closer to the actual area under the curve.
4. The method of Riemann sums is used to find the area by using left endpoints, right endpoints, or midpoints to determine rectangle heights.
5. In practical applications, using rectangles to estimate areas helps in understanding concepts of integration and provides a foundation for more complex calculations.

Review Questions

• How does increasing the number of rectangles used to approximate an area affect the accuracy of that approximation?
  • Increasing the number of rectangles used for approximation reduces their width, leading to a more accurate representation of the area between curves. As rectangles become narrower, they better conform to the shape of the curves. This approach aligns with the concept of taking limits, where as the number of rectangles approaches infinity, the approximation converges to the exact area under the curve.
• In what ways do left endpoint and right endpoint Riemann sums differ when estimating areas using rectangles?
  • Left endpoint and right endpoint Riemann sums differ in how they select heights for each rectangle. The left endpoint sum uses the value of the function at the left edge of each subinterval, while the right endpoint sum uses the value at the right edge. This can lead to different estimates; for increasing functions, left sums typically underestimate and right sums overestimate. Conversely, for decreasing functions, left sums will overestimate and right sums will underestimate.
• Evaluate how using upper and lower sums provides insights into determining definite integrals when calculating areas between curves.
  • Using upper and lower sums offers a way to establish bounds on definite integrals by approximating areas from above and below. The upper sum uses maximum values within subintervals for height, while the lower sum uses minimum values. This method not only highlights how close these estimates come to actual integral values but also reinforces understanding of limits in calculus; as more rectangles are added, both bounds converge towards the true area between curves, demonstrating fundamental principles behind integration.

Related terms

Riemann Sum: A method of approximating the total area under a curve by dividing it into smaller segments and summing the areas of rectangles.

Definite Integral: A mathematical concept that calculates the exact area under a curve over a specified interval using limits.

Upper and Lower Sums: Approximations of the area under a curve that use the maximum and minimum function values within subintervals to create upper and lower estimates of the area.