Answer the following True or False. For all 1

Question

Answer the following True or False. For all  1 < a < b,int_(a)^(b)x^(2)dx > int_(a)^(b)xdx. True False

Bytelearn · Accepted Answer

Calculate Integral of x^2: Let's first calculate the integral of x^2 from a to b. The antiderivative of x^2 is (1/3)x^3. So, the definite integral from a to b is (1/3)b^3 - (1/3)a^3.Calculate Integral of x: Now, let's calculate the integral of x from a to b. The antiderivative of x is (1/2)x^2. So, the definite integral from a to b is (1/2)b^2 - (1/2)a^2.Comparison of Integrals: We need to compare (1/3)b^3 - (1/3)a^3 with (1/2)b^2 - (1/2)a^2. Since 1 < a < b, we know that b^3 > b^2 and a^3 > a^2. Therefore, (1/3)b^3 > (1/2)b^2 and (1/3)a^3 > (1/2)a^2.Consider Coefficients: However, we need to be careful with the comparison because the coefficients (1/3) and (1/2) affect the values. We can't directly conclude that (1/3)b^3 - (1/3)a^3 is greater than (1/2)b^2 - (1/2)a^2 just because b^3 > b^2 and a^3 > a^2. We need to consider the entire expression.Analyzing Expressions: Let's analyze the expressions further. For b^3 to be greater than b^2, b must be greater than 1, which is given. Similarly, for a^3 to be greater than a^2, a must be greater than 1, which is also given.Compare Coefficients: Now, let's compare the coefficients. (1/3) is less than (1/2), which means that for the same base, the term with the coefficient (1/3) will be smaller than the term with the coefficient (1/2). This means that we need to be cautious in our comparison because the larger exponent is paired with the smaller coefficient.Behavior of Functions: To make a proper comparison, we can consider the behavior of the functions x^2 and x as x increases. The function x^2 grows faster than the function x as x increases beyond 1. This means that the area under the curve of x^2 from a to b will grow more rapidly than the area under the curve of x from a to b.Conclusion: Since the growth rate of x^2 is faster than that of x, and both a and b are greater than 1, the integral of x^2 from a to b will indeed be greater than the integral of x from a to b. This means that the original statement is true.

Answer the following True or False. For all 1 < a < b, \int_{a}^{b} x^{2} \, dx > \int_{a}^{b} x \, dx. $\newline$ True $\newline$ False

Full solution

More problems from Properties of matrices

Related topics

Answer the following True or False. For all 1 < a < b, \int_{a}^{b} x^{2} \, dx > \int_{a}^{b} x \, dx. \newlineTrue \newlineFalse

Answer the following True or False. For all 1 < a < b, \int_{a}^{b} x^{2} \, dx > \int_{a}^{b} x \, dx. $\newline$ True $\newline$ False