Question:medium
Which ONE of the following options is the closest approximation to the value of the given definite integral?
\[ \int_0^1 e^{-x^2} dx \]
Note: Use numerical integration
Which ONE of the following options is the closest approximation to the value of the given definite integral?
\[ \int_0^1 e^{-x^2} dx \]
Note: Use numerical integration
\[ \int_0^1 e^{-x^2} dx \]
Note: Use numerical integration
Show Hint
e^-x^2 has no simple antiderivative. Use the trapezoidal rule with f(0)=1 and f(1)=1/e over the single interval [0,1].
Updated On: Aug 5, 2026
- \( 1 \)
- \( \frac{1}{e} \)
- \( \frac{e+1}{2e} \)
- \( \frac{1+e}{2} \)
Show Solution
The Correct Option is C
Solution and Explanation
Step 1: Get a rough decimal value of the integral first.
The true value of $\int_0^1 e^{-x^2} dx$ is close to $0.7468$, a well known constant tied to the error function.
We can use this rough number to test which option lands closest, without needing the exact trapezoidal formula first.
Step 2: Convert each option into a decimal.
$\text{Option A: } 1 = 1.000$
$\text{Option B: } \frac{1}{e} \approx 0.368$
$\text{Option C: } \frac{e+1}{2e} \approx \frac{3.718}{5.437} \approx 0.684$
$\text{Option D: } \frac{1+e}{2} \approx \frac{3.718}{2} \approx 1.859$
Step 3: Compare each decimal to the true value 0.7468.
Option A is too high, option B is too low, option D is far too high, but option C at about 0.684 sits closest to the true value among all four choices.
Step 4: Confirm this using the trapezoidal shortcut.
For a curve that dips down like $e^{-x^2}$, the trapezoidal estimate using just the two end points, $f(0)=1$ and $f(1)=1/e$, gives exactly $\frac{1+1/e}{2} = \frac{e+1}{2e}$.
This matches option C again, so both the decimal comparison and the formula route agree on the same choice.
Step 5: Note why this is only an approximation.
The trapezoidal rule with just one segment slightly overestimates the true area here, since it connects the two end points with a straight line while the real curve bends below that line.
Even so, among the four given options, it is still the closest match to the actual integral value.
Final Answer:
Comparing decimal values and using the trapezoidal rule both point to option C as the closest approximation. \[ \boxed{\frac{e+1}{2e}} \]
The true value of $\int_0^1 e^{-x^2} dx$ is close to $0.7468$, a well known constant tied to the error function.
We can use this rough number to test which option lands closest, without needing the exact trapezoidal formula first.
Step 2: Convert each option into a decimal.
$\text{Option A: } 1 = 1.000$
$\text{Option B: } \frac{1}{e} \approx 0.368$
$\text{Option C: } \frac{e+1}{2e} \approx \frac{3.718}{5.437} \approx 0.684$
$\text{Option D: } \frac{1+e}{2} \approx \frac{3.718}{2} \approx 1.859$
Step 3: Compare each decimal to the true value 0.7468.
Option A is too high, option B is too low, option D is far too high, but option C at about 0.684 sits closest to the true value among all four choices.
Step 4: Confirm this using the trapezoidal shortcut.
For a curve that dips down like $e^{-x^2}$, the trapezoidal estimate using just the two end points, $f(0)=1$ and $f(1)=1/e$, gives exactly $\frac{1+1/e}{2} = \frac{e+1}{2e}$.
This matches option C again, so both the decimal comparison and the formula route agree on the same choice.
Step 5: Note why this is only an approximation.
The trapezoidal rule with just one segment slightly overestimates the true area here, since it connects the two end points with a straight line while the real curve bends below that line.
Even so, among the four given options, it is still the closest match to the actual integral value.
Final Answer:
Comparing decimal values and using the trapezoidal rule both point to option C as the closest approximation. \[ \boxed{\frac{e+1}{2e}} \]
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