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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
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