What is the trapezoid rule ?

The trapezoid rule is a numerical method used to find the approximate area enclosed by a curve , the x -axis and two vertical lines .

True or False? The trapezoid rule uses rectangles to approximate the area under a curve.

False. The trapezoid rule uses trapezoids to approximate the area under a curve.

True or False? The trapezoid rule always overestimates the true area under a curve.

False. The trapezoid rule can either overestimate or underestimate the true area under a curve, depending on the shape of the curve.

True or False? The trapezoid rule becomes more accurate as the number of trapezoids decreases .

False. The trapezoid rule becomes more accurate as the number of trapezoids increases .

What is integration ?

Integration is the opposite (inverse) of differentiation . It is the process of finding the expression of a function ( antiderivative ) from an expression of the function's derivative (gradient function).

True or False? When integrating a power of x , the power is decreased by 1.

False. When integrating a power of x , the power is increased by 1. (The power is decreased by 1 when differentiating powers of x .)

How do you integrate a sum or difference of powers of x ?

To integrate a sum or difference of powers of x , integrate each term individually using the power rule and combine the results. I.e. the integral of a sum or difference of terms is equal to the sum or difference of the integrals of the individual terms.

What is a definite integral ?

A definite integral is an integral with specified upper and lower limits , used to calculate the exact area under a curve .

True or False? The constant of integration is needed in definite integration.

False. The constant of integration is not needed in definite integration. (If it were included, it would just cancel out.)

How can you find the constant of integration if a point on the curve is known?

To find the constant of integration when a point on the curve is known: substitute the x- and y -coordinates of the point into the general antiderivative equation, then solve for the constant of integration C .

True or False? Modern graphic calculators ( GDCs ) will always produce exact answers for definite integrals.

False. Modern graphic calculators ( GDCs ) may not always produce exact answers for definite integrals, and care should be taken when interpreting results.

What information is needed to evaluate a definite integral using a GDC ?

To evaluate a definite integral using a GDC , you need: the function to be integrated (integrand), the lower integration limit , and the upper integration limit .

How can you use a GDC to find the area under a curve graphically ?

To find the area under a curve graphically using a GDC: Plot the function. Select the area or integral option. Input or select the lower and upper limits on the graph.

True or False? When finding the area under a straight line , the only possible method is to use integration .

False. Definite integration can find the area under a line (and integration may be preferred for consistency with other area calculations). Just use the equation of the line as the 'equation of the curve' in the integral. However the area under a straight line will be either a right triangle or a trapezoid ( trapezium ). It may therefore be easier to use the area formulae for those 2D shapes to work out the area.

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FrontNumerical Integration using the Trapezoid Rule
What is the trapezoid rule?
FrontNumerical Integration using the Trapezoid Rule
What does $h$ represent in the trapezoid rule formula?
BackNumerical Integration using the Trapezoid Rule
In the trapezoid rule formula, $h$ represents the width of each trapezoid
This is calculated by $h = \frac{b - a}{n}$
Where:
$a$ and $b$ are the integration limits
$n$ is the number of trapezoids

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Cards in this collection (27)

What is the trapezoid rule?
The trapezoid rule is a numerical method used to find the approximate area enclosed by a curve, the x-axis and two vertical lines.
True or False?
The trapezoid rule uses rectangles to approximate the area under a curve.
False.
The trapezoid rule uses trapezoids to approximate the area under a curve.
What does $h$ represent in the trapezoid rule formula?
In the trapezoid rule formula, $h$ represents the width of each trapezoid
This is calculated by $h = \frac{b - a}{n}$
Where:
- $a$ and $b$ are the integration limits
- $n$ is the number of trapezoids
State the formula for the trapezoid rule.
The trapezoid rule formula is $\int_{a}^{b} y d x \approx \frac{1}{2} h ((y_{0} + y_{n}) + 2 (y_{1} + y_{2} + . . . + y_{n - 1}))$
Where:
- $a$ and $b$ are the integration limits
- $n$ is the number of trapezoids
- $h = \frac{b - a}{n}$ is the width of the trapezoids
- $y_{0}, y_{1}, . . ., y_{n}$ are the y-coordinates of the vertices at the top of the trapezoids
This equation is in the exam formula booklet.
True or False?
The trapezoid rule always overestimates the true area under a curve.
False.
The trapezoid rule can either overestimate or underestimate the true area under a curve, depending on the shape of the curve.
True or False?
The trapezoid rule becomes more accurate as the number of trapezoids decreases.
False.
The trapezoid rule becomes more accurate as the number of trapezoids increases.
What is integration?
Integration is the opposite (inverse) of differentiation.
It is the process of finding the expression of a function (antiderivative) from an expression of the function's derivative (gradient function).
What does the symbol $\int$ mean?
The symbol $\int$ means "integrate".
E.g. $\int y d x$ means integrate (or 'find the integral of') $y$ with respect to $x$
State the formula for integrating powers of x.
The formula for integrating powers of x is $\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C, n \neq - 1$
Where:
- $C$ is the constant of integration
This formula is in the exam formula booklet.
What is the constant of integration?
The constant of integration is a constant term that appears in every indefinite integral, representing the fact that a function $f (x)$ has infinitely many antiderivatives.
The constant of integration is usually written as ' $+ C$ ' at the end of the indefinite integral.
True or False?
When integrating $a x^{n}$ , the result for $x^{n}$ is multiplied by the constant $a$ .
True.
When integrating $a x^{n}$ , the result for $x^{n}$ is multiplied by the constant $a$ .
What is the integral of a constant $a$ ?
The integral of a constant $a$ is $a x$ ( $+ C$ if it is an indefinite integral).
True or False?
When integrating a power of x, the power is decreased by 1.
False.
When integrating a power of x, the power is increased by 1.
(The power is decreased by 1 when differentiating powers of x.)
True or False?
The formula for integrating $x^{n}$ is valid for all possible values of $n$ .
False.
The formula for integrating $x^{n}$ is not valid for $n = - 1$ .
It is valid for all other values of $n$ .
How do you integrate a sum or difference of powers of x?
To integrate a sum or difference of powers of x, integrate each term individually using the power rule and combine the results.
I.e. the integral of a sum or difference of terms is equal to the sum or difference of the integrals of the individual terms.
True or False?
To find the integral of a product like $3 x (4 x - 7)$ , find the integrals of the individual factors ( $3 x$ and $4 x - 7$ ) and multiply them together.
False.
To find the integral of a product like $3 x (4 x - 7)$ , you first need to expand the brackets to get $12 x^{2} - 21 x$ . Then that can be integrated as usual using the powers of $x$ formula.
Similarly, you cannot find the integral of a quotient (fraction) by finding the integrals of the numerator and denominator and then dividing.
What is a definite integral?
A definite integral is an integral with specified upper and lower limits, used to calculate the exact area under a curve.
True or False?
The constant of integration is needed in definite integration.
False.
The constant of integration is not needed in definite integration.
(If it were included, it would just cancel out.)
What is meant by the area under a curve?
The area under a curve refers to the area bounded by:
- the graph of $y = f (x)$ ,
- the x-axis,
- and two vertical lines $x = a$ and $x = b$ .
What are the limits of integration?
Limits of integration (or integration limits) are the upper and lower bounds of a definite integral, determining the interval over which the integration is performed.
E.g. in the definite integral $\int_{a}^{b} y d x$
- $a$ is the lower integration limit
- $b$ is the upper integration limit
True or False?
The y-axis can be one of the boundaries when finding the area under a curve.
True.
The y-axis can be one of the boundaries when finding the area under a curve
In this case the corresponding integration limit would be $x = 0$ .
How can you find the constant of integration if a point on the curve is known?
To find the constant of integration when a point on the curve is known:
- substitute the x- and y-coordinates of the point into the general antiderivative equation,
- then solve for the constant of integration C.
True or False?
Modern graphic calculators (GDCs) will always produce exact answers for definite integrals.
False.
Modern graphic calculators (GDCs) may not always produce exact answers for definite integrals, and care should be taken when interpreting results.
What information is needed to evaluate a definite integral using a GDC?
To evaluate a definite integral using a GDC, you need:
- the function to be integrated (integrand),
- the lower integration limit,
- and the upper integration limit.
How can you use a GDC to find the area under a curve graphically?
To find the area under a curve graphically using a GDC:
- Plot the function.
- Select the area or integral option.
- Input or select the lower and upper limits on the graph.
What is the significance of roots of the equation $f (x) = 0$ when finding limits for calculating an area under a curve?
Roots of the equation $f (x) = 0$ represent the x-coordinates of the point(s) where the graph of $y = f (x)$ crosses the x-axis.
These x-coordinates will sometimes be used as limits when finding an area under a curve.
True or False?
When finding the area under a straight line, the only possible method is to use integration.
False.
Definite integration can find the area under a line (and integration may be preferred for consistency with other area calculations). Just use the equation of the line as the 'equation of the curve' in the integral.
However the area under a straight line will be either a right triangle or a trapezoid (trapezium). It may therefore be easier to use the area formulae for those 2D shapes to work out the area.