9.4.1 Approximating Areas of Plane Regions

Question 1

Approximating Areas of Plane Regions

Answer 1

• The two key questions of calculus have a subtle connection.
• When trying to find the area of a complicated region, try approximating the area with rectangles. As the number of rectangles increases, the approximation becomes more accurate.

Question 2

note

Answer 2

• The two big questions in calculus are “How do you find
  instantaneous velocity?” and “How do you find the area of exotic shapes?”
• Consider an exotic shape defined by the graph of a function, the x-axis, and two points on the axis.
• Since calculating the area of the region might be difficult, you could approximate the area by dividing the region into rectangles. The area covered by the rectangles can be expressed as the sum of the areas of the individual rectangles. Sigma ( ) notation provides a shorthand expression for the sum.
• You can improve the approximation of the area by increasing the number of rectangles. With more rectangles, less of the region is left uncovered.
• Notice that the base of each rectangle is thinner than with fewer rectangles. As the number of rectangles increases, the lengths of the bases will approach zero.
• To completely cover the region, you will need infinitely many rectangles whose bases are infinitesimal. The sum of their areas will equal the area of the region.
• In other words, you need to take the limit of the areas of the rectangles as the lengths of their bases approach zero.

Question 3

Approximate the area under the curve y = −x^ 2 + 16 between 0 and 4 by finding the area of the circumscribed rectangles using 4 subdivisions.

Answer 3

A ≈ 50

Question 4

Which of the following approximations is closest to the actual area underneath this curve between a and b ?

Answer 4

-

Question 5

When using rectangular partitions to approximate the area under a curve, how does increasing the number of partitions (or rectangles) used to approximate an area generally affect the approximation?

Answer 5

Increasing the number of partitions or rectangles generally generates a smaller error in the approximation.