Class Notes (1,100,000)

US (460,000)

UC-Irvine (10,000)

MATH (1,000)

MATH 2B (600)

ERJAEE, G. (20)

Lecture 2

Department

MathematicsCourse Code

MATH 2BProfessor

ERJAEE, G.Lecture

2This

**preview**shows half of the first page. to view the full**3 pages of the document.** MATH 2B - Lecture 2 - Areas and Distances

Area of a Rectangle

● The starting point of integral calculus is the problem of calculating area

. The

naÏve concept of area comes from the formula for the area of rectangle

:

Area = Length · Width

● The area of a triangle is then immediately half that of a rectangle, and any shape

which may be subdivided into triangles may have its area computed.

Area Under the Curve

● The primary question of integral calculus is of how to extend this idea to cover

shapes which cannot be built from triangles.

● For example, below is the graph of the curve between 0 ≤ x ≤ 2;)xy = 1 + x−3

13

how are we to compute the shaded area?

● We approximate the area under the curve by rectangles and sum the areas of

these. If we take a larger number of rectangles, we will hopefully obtain a better

approximation to the desired area

○ Does this approximating process work for all functions?

○ Does it matter how

we choose the rectangles

● The process of this is inefficient because a

human can’t evaluate the sum of 1500 area of

rectangles

Upper and Lower Bounds for Areas Example

● The area under the curve between x = 0y=x3

and x = 1 is estimated using rectangles

● Upper bound: n rectangles fitting just over curve

###### You're Reading a Preview

Unlock to view full version

Subscribers Only