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Estimation using rectangles

Estimate the area between the x -axis and the graph of f ( x ) = x 2 + 1 over the interval [ 0 , 3 ] by using the three rectangles shown in [link] .

A graph of the parabola f(x) – x^2 + 1 drawn on graph paper with all units shown. The rectangles completely contained under the function and above the x-axis in the interval [0,3] are shaded. This strategy sets the heights of the rectangles as the smaller of the two corners that could intersect with the function. As such, the rectangles are shorter than the height of the function.
The area of the region under the curve of f ( x ) = x 2 + 1 can be estimated using rectangles.

The areas of the three rectangles are 1 unit 2 , 2 unit 2 , and 5 unit 2 . Using these rectangles, our area estimate is 8 unit 2 .

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Estimate the area between the x -axis and the graph of f ( x ) = x 2 + 1 over the interval [ 0 , 3 ] by using the three rectangles shown here:

A graph of the same parabola f(x) = x^2 + 1, but with a different shading strategy over the interval [0,3]. This time, the shaded rectangles are given the height of the taller corner that could intersect with the function. As such, the rectangles go higher than the height of the function.

16 unit 2

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Other aspects of calculus

So far, we have studied functions of one variable only. Such functions can be represented visually using graphs in two dimensions; however, there is no good reason to restrict our investigation to two dimensions. Suppose, for example, that instead of determining the velocity of an object moving along a coordinate axis, we want to determine the velocity of a rock fired from a catapult at a given time, or of an airplane moving in three dimensions. We might want to graph real-value functions of two variables or determine volumes of solids of the type shown in [link] . These are only a few of the types of questions that can be asked and answered using multivariable calculus    . Informally, multivariable calculus can be characterized as the study of the calculus of functions of two or more variables. However, before exploring these and other ideas, we must first lay a foundation for the study of calculus in one variable by exploring the concept of a limit.

A diagram in three dimensional space, over the x, y, and z axis where z = f(x,y). The base is the x,y axis, and the height is the z axis. The base is a rectangle contained in the x,y axis plane. The top is a surface of changing height with corners located directly above those of the rectangle in the x,y plane.. The highest point is above the corner at x=0, y=0. The lowest point is at the corner somewhere in the first quadrant of the x, y plane. The other two points are roughly the same height and located above the corners on the x axis and y axis. Lines are drawn connecting the corners of the rectangle to those of the surface.
We can use multivariable calculus to find the volume between a surface defined by a function of two variables and a plane.

Key concepts

  • Differential calculus arose from trying to solve the problem of determining the slope of a line tangent to a curve at a point. The slope of the tangent line indicates the rate of change of the function, also called the derivative . Calculating a derivative requires finding a limit.
  • Integral calculus arose from trying to solve the problem of finding the area of a region between the graph of a function and the x -axis. We can approximate the area by dividing it into thin rectangles and summing the areas of these rectangles. This summation leads to the value of a function called the integral . The integral is also calculated by finding a limit and, in fact, is related to the derivative of a function.
  • Multivariable calculus enables us to solve problems in three-dimensional space, including determining motion in space and finding volumes of solids.

Key equations

  • Slope of a Secant Line
    m sec = f ( x ) f ( a ) x a
  • Average Velocity over Interval [ a , t ]
    v ave = s ( t ) s ( a ) t a

For the following exercises, points P ( 1 , 2 ) and Q ( x , y ) are on the graph of the function f ( x ) = x 2 + 1 .

[T] Complete the following table with the appropriate values: y -coordinate of Q , the point Q ( x , y ) , and the slope of the secant line passing through points P and Q . Round your answer to eight significant digits.

x y Q ( x , y ) m sec
1.1 a. e. i.
1.01 b. f. j.
1.001 c. g. k.
1.0001 d. h. l.

a. 2.2100000; b. 2.0201000; c. 2.0020010; d. 2.0002000; e. (1.1000000, 2.2100000); f. (1.0100000, 2.0201000); g. (1.0010000, 2.0020010); h. (1.0001000, 2.0002000); i. 2.1000000; j. 2.0100000; k. 2.0010000; l. 2.0001000

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Questions & Answers

A Function F(X)=Sinx+cosx is odd or even?
f(x)=1/1+x^2 |=[-3,1]
Yuliana Reply
apa itu?
determine the area of the region enclosed by x²+y=1,2x-y+4=0
Gerald Reply
Hi too
hello please anyone with calculus PDF should share
Which kind of pdf do you want bro?
can I get calculus in pdf
How to use it to slove fraction
Tricia Reply
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Im not in college but this will still help
how can we scatch a parabola graph
Dever Reply
how can I solve differentiation?
Sir Reply
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
For example any questions...
what is the procedures in solving number 1?
Vier Reply
review of funtion role?
Md Reply
for the function f(x)={x^2-7x+104 x<=7 7x+55 x>7' does limx7 f(x) exist?
find dy÷dx (y^2+2 sec)^2=4(x+1)^2
Rana Reply
Integral of e^x/(1+e^2x)tan^-1 (e^x)
naveen Reply
why might we use the shell method instead of slicing
Madni Reply
albert Reply
find the values of c such that the graph of f(x)=x^4+2x^3+cx^2+2x+2
Ramya Reply
anyone to explain some basic in calculus
Adegoke Reply
I can
Practice Key Terms 8

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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