University:
Felician University
Course:
MATH-002 | Basic Math Skills-Algebra
Academic year:

2022
Views:

431

Pages:

4
Author:

Clint W.

• CALCULUS ТНЕ TRAPEZOIDAL RULE (4.3) NOTES ASSIGNMENT: Web Assign Objectives: • Approximate а definite integral using the Trapezoidal Rule. • Analyze the approximate errors in the Trapezoidal Rule. The Trapezoidal Rule Some elementary functions simp\y do not have antiderivatives that аге elementary functions. When you need to evaluate а definite integral involving а function whose antiderivative cannot Ье found, you can use an approximation technique. One way to approximate а definite integral is to use п trapezoids, as shown in Figure 4.28. ln the development of this method, assume thatf is continuous and positive on the interval [а, Ь]. So, the definite integral (" Ja .f(x) с/х represents the агеа of the region bounded Ьу the graph off and the x-axis, from х = а to х = Ь. First, partition the interval [а, Ь] into п sublntervals, each of width Лх = (Ь- а)/п, such that а= Хо< Х1 < Х2 < • • • < Хп = Ь. Then form а trapezoid for each sublnterval. (See Figure 4.29.) The агеа of the i th trapezoid is Area of ith trapezoid = [J(хн)/ f(x;) ](11 � а). � The агеа of the region can Ье approximated using four trapezoids. i This implies that the sum of the areas of the п trapezoids is ,, • The result is summarized in the following theorem. THEOREM 4.9 The Trapezoidal Rule Let f Ье contiпuous оп [а, Ь]. Tl1e T1·apezoidal RuJe J:' J(x) dx = Ь � а [I(x0) f01· appгoxiшating J:: f(x) clx is + 2/(х,) + 2/(х") + • • • + 2f(x,,_ ,) + .f(x,,)]. Moreover, as п---1 оо, the !'ight-l1a11d side appmacl1es I:: .f(x) dx. Notice in the next example that you can predict that the Trapezoidal Rule will underestimate the integral Ьу looking at the concavity of the graph. When the graph of а function is concave upward on а given interva\, the Trapezoidal Rule . � will overestimate the value of the integral over the interval. When the graph is concave downward, as in Example 1, the rule will underestimate the value of the integral.