Trapezoidal Approximation Objective: To find area using trapezoids.

Trapezoidal Approximation

Objective: To find area using trapezoids.

Trapezoidal Approximation

• We will now approximate the area under a curve by using trapezoids rather than rectangles. This is only an approximation; we will never take the limit to find the exact area.

Trapezoidal Approximation

• As we did before, we will draw vertical lines to divide the interval into n subintervals. This time, we will construct n trapezoids.

Trapezoidal Approximation

• As we did before, we will draw vertical lines to divide the interval into n subintervals. This time, we will construct n trapezoids.

• The area of a trapezoid is . The height of the each trapezoid is what we called before.

)( 212 bbh x

Trapezoidal Approximation

• The area of the first trapezoid is

• The area of the second trapezoid is

• The area of the third trapezoid is

)( 102)( yynab

)( 212)( yynab

)( 322)( yynab

Trapezoidal Approximation

• The area of the first trapezoid is

• The area of the second trapezoid is

• This leads us to the following:

)( 102)( yynab

)( 212)( yynab

Sample AP Question

Sample AP Question

B500.3230)...18(2)5.19(220102010

Note

• The AP book made the following notes:• DO NOT use your calculator’s statistics regression

equation to find f(x) and then the Fundamental Theorem of Calculus; this may give you a more accurate answer (D) but it is not what you were asked for. The left hand sum is (E) and the right hand sum is (A).

AP Question

• Using the subintervals [1,5], [5,8], and [8,10], what is the trapezoidal approximation to

10

1

)cos2( dxx

AP Question

• Using the subintervals [1,5], [5,8], and [8,10], what is the trapezoidal approximation to

10

1

)cos2( dxx

AP Question

• Using the subintervals [1,5], [5,8], and [8,10], what is the trapezoidal approximation to

10

1

)cos2( dxx

AP Question

• The expression is the trapezoidal approximation for which of the following definite integrals?

a) b) c)

d) e)

54/132224/52141

dxx3

1

dxx5

1

dxx 4

0

2 1

dxx 2

0

2 1 dxx

2

1

2 1

AP Question

• The expression is the trapezoidal approximation for which of the following definite integrals?

• There are 4 trapezoids (why), so n = 4.• We know that and if n = 4 it becomes so b – a = 2. Our only two choices are A and D.

54/132224/52141

n

ab

24

1 84

1 ab

AP Question

• The expression is the trapezoidal approximation for which of the following definite integrals?

• There are 4 trapezoids (why), so n = 4.• We know that and if n = 4 it becomes so b – a = 2. Our only two choices are A and D.• If a = 0, the bases are f(0), f(1/2), f(1), f(3/2), f(2).• If a = 1, the bases are f(1), f(3/2), f(2), f(5/2), f(3).

54/132224/52141

n

ab

24

1 84

1 ab

AP Question

• The expression is the trapezoidal approximation for which of the following definite integrals?

• There are 4 trapezoids (why), so n = 4.• We know that and if n = 4 it becomes so b – a = 2. Our only two choices are A and D.• If a = 0, the bases are f(0), f(1/2), f(1), f(3/2), f(2).• If a = 1, the bases are f(1), f(3/2), f(2), f(5/2), f(3).

54/132224/52141

n

ab

24

1 84

1 ab

2)(

)(

1)(

2

45

1

0

xf

xf

xf

5)(

)(

4

413

3

xf

xf

AP Question

• The expression is the trapezoidal approximation for which of the following definite integrals?

a) b) c)

d) e)

54/132224/52141

dxx3

1

dxx5

1

dxx 4

0

2 1

dxx 2

0

2 1 dxx

2

1

2 1 5)(

)(

4

413

3

xf

xf

2)(

)(

1)(

2

45

1

0

xf

xf

xf

AP Question

• The expression is the trapezoidal approximation for which of the following definite integrals?

a) b) c)

d) e)

54/132224/52141

dxx3

1

dxx5

1

dxx 4

0

2 1

dxx 2

0

2 1 dxx

2

1

2 1 2)(

)(

2

45

1

xf

xf

5)(

)(

4

413

3

xf

xf