the trapezoidal rule for approximating Definite Integrals

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本文介绍使用等分区间法近似计算积分的方法，通过实例演示如何将积分区间分为多个子区间，进而估算两个特定积分的值。第一个积分是从0到1/2的1/(1+x^2)dx，使用五个子区间得到近似值0.463；第二个积分是从0到3的sqrt(1+x^3)dx，使用六个子区间得到近似值7.39。

Approximate $∫012dx1+x2\int_{0}^\frac{1}{2} \frac{d x}{1+x^{2}}$ by using five subintervals. $\quad$ Ans. 0.463

表达式产出：

leftPoint = 0
rightPoint = 1 / 2
subIntervalNum = 5
subIntervalLen = '{}/{}'.format(rightPoint - leftPoint, subIntervalNum)

expressionStrList = []
for i in range(subIntervalNum + 1):
    expressionStrList.append('1/(1+({leftPoint} + {subIntervalLen} * {i})^2)'.format(leftPoint=leftPoint, subIntervalLen=subIntervalLen, i=i))
joinedExpressionStr = ' + '.join(expressionStrList)
print(joinedExpressionStr)

print('{subIntervalLen}*(1/2*{joinedExpressionStr}*1/2)'.format(subIntervalLen=subIntervalLen, joinedExpressionStr=joinedExpressionStr))

Approximate $∫031+x3dx\int_{0}^{3} \sqrt{1+x^{3}} d x$ using six subintervals. $\quad$ Ans. $7.39 .$

leftPoint = 0
rightPoint = 3
subIntervalNum = 6
subIntervalLen = '{}/{}'.format(rightPoint - leftPoint, subIntervalNum)

expressionStrList = []
for i in range(subIntervalNum + 1):
    expressionStrList.append('sqrt(1+({leftPoint} + {subIntervalLen} * {i})^3)'.format(leftPoint=leftPoint, subIntervalLen=subIntervalLen, i=i))
joinedExpressionStr = ' + '.join(expressionStrList)
print(joinedExpressionStr)

print('{subIntervalLen}*(1/2*{joinedExpressionStr}*1/2)'.format(subIntervalLen=subIntervalLen, joinedExpressionStr=joinedExpressionStr))