NUMERICAL SECTION DERIVATIVES r +2x dr 10. Answer the following True or False qu

Question

NUMERICAL SECTION DERIVATIVES r +2x dr 10. Answer the following True or False questions about approximating the area between the function and the z axis (a) A smaller ?z will give a more accurate approximation (b) Right Side Rectangles are always less accurate than Left Side Rectangles (c) Midpoint Rectangles always have the least error (d) As ?? goes to 0 the Area calculation equals the Integral value (e) The Summation of Rectangles is easier and quicker to use than the Fundamental Theorem of Calculus to evaluate the integral.

Explanation / Answer

(a) The statement is TRUE

Reason: Since as we decrease the value of Delta x and make it more and more closer to 0, then the appeoximation will reach more and more closer towards the value obtained by integration (which is correct area). Hence the smaller Delta x will give a more accurate approximation

(b) The statement is FALSE

Reason: Let us assume the function which is exponentially growing then the left hand approximation is way more less than the actual value, and in this case right hand approximation will be better than the left hand approximation and more closer to the result

(c) The statement is FALSE

Reason: In most of the cases, the mid-point approximation is better than the left and right hand estimates. But for specific cases, the left and right hand estimates outperforms the mid-point approximation

(d) The statement is TRUE

According to the fundamental theorem of integration, the value of Detla x considered should be tending to zero, which means we have done infinite paritions of the region space resulting in the approximate area

(e) The statement is FALSE

Reason: The fundamental theorem of integration is an easier method, since we just need to calculate the integral and substitute the limits. The summation of rectangle is more error prone and secondly as the value of Delta x becomes small, the number of rectangles increases making the calculation more difficult and time consuming

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