Geometry problems Use a table of integrals to solve the...

Key Concepts

Completing the Square

Completing the square is a technique used to convert a quadratic expression into a perfect square plus or minus a constant. This simplifies the expression under a square root or in a denominator, making it easier to match with standard integral forms or to apply trigonometric substitution during integration.

Standard Integrals

Standard integrals refer to a collection of common antiderivatives that are often tabulated in integral tables. By recognizing parts of an integrand that fit these common forms, one can directly evaluate integrals without resorting to more complicated, ad?hoc methods.

Trigonometric Substitution

Trigonometric substitution is a method used for integrating expressions involving square roots of quadratic polynomials. By substituting variables with appropriate trigonometric functions, the integral can be simplified using trigonometric identities, allowing for easier evaluation.

Definite Integrals and Area Under a Curve

Definite integrals are used to find the accumulated area under a curve within given bounds. When the function is non-negative over an interval, the definite integral directly represents the area between the function and the x-axis, which is a key concept in solving area problems in geometry.

Transcript

00:01 Section 7 .6, problem number 44.

00:03 So here i'm asked to find the area between a curve, which you see given here, 1 over the square root of x squared minus 2x plus 2.

00:11 Area between that curve, the x -axis, when x goes from 0 to 3.

00:16 So let's take a look at it first.

00:18 So what you see here is the graph of the initial curve, which you see here.

00:24 I've got x goes from 0 to 3.

00:28 And so i'm being asked to find this shape.

00:30 Data area, which means basically i need to find the integral from 0 to 3 for this function.

00:37 And it turns out to be about 2 .325.

00:41 So i'm going to get an exact value, and its approximation should be very close to that number.

00:46 So in this case, what i'm doing, i'm using integral lookup tables.

00:52 And so let's just take a look.

00:54 I've been using the table that you see here that's referenced.

00:58 And so i go and i look.

01:00 Look one up and this just illustrates you got to sort of be on top of your game here there's a typo in this particular problem so you see this actually what this one is is this is the integral of the one that i'm looking for one over a x squared plus bx plus c dx they forgot to put the square root sign and this works when a is greater than zero so this is actually what i'm looking for.

01:28 So all the key reason that you're going to have to really know how to integrate, you can't rely always on an integral table to work it for you.

01:35 So let's go back and let's just write down what that solution would be.

01:41 So according to our integral lookup table, i know that the indefinite integral 1 over square root a x squared plus bx plus c dx is going to be equal to 1 .1 .1 .2.

01:59 It is going to be equal to 1 over the square root of a, natural log 2ax plus b plus 2 square root of a, and then the square root of ax squared plus bx plus c, and then all of this plus a constant, and this is when a is greater than 0.

02:27 So in my case, i need to do this particular problem where a, coefficient of the quadratic term is 1, b, coefficient of the linear term, is negative 2, and c, the constant term is positive 2.

02:44 So i will need to substitute those values in, so just using this formula, you will have 1 over the square root of 1, natural log, 2 times 1, times x.

02:58 Plus negative 2 plus 2 times 1 and then the square root of x squared minus 2x plus 2 and then all of this is evaluated from 0 to 3.

03:20 So let's clean this up just a little bit.

03:24 This is the natural log of 2x minus 2 plus 2, square, root of x squared minus 2x plus 2 evaluate from 0 to 3...

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